Pythagorean Theorem Worksheets

Related Topics: More Math Worksheets More Printable Math Worksheets 7th Grade Math

There are five sets of triangle worksheets:

  • Triangle Sum Theorem
  • Exterior Angle Theorem
  • Area of Triangle

Area of Triangle using Sine

  • Pythagorean Theorem

Examples, solutions, videos, and worksheets to help grade 7 students learn how to use the Pythagorean Theorem.

How to use the Pythagorean Theorem?

There are five sets of Pythagorean Theorem worksheets:

  • Find the length of the hypotenuse
  • Find the length of a side
  • Find the length in two steps
  • Check for right triangle
  • Pythagorean Theorem Word Problems

The Pythagorean Theorem also called the Pythagoras’ Theorem is a fundamental principle in geometry that relates the sides of a right triangle.

It states that in a right triangle (a triangle with one 90-degree angle), the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Mathematically, the Pythagorean Theorem can be expressed as: c² = a² + b²

  • c represents the length of the hypotenuse.
  • a and b represent the lengths of the other two sides.

The Pythagorean Theorem is widely used to calculate the length of any one side of a right triangle when the lengths of the other two sides are known. It is an essential tool in geometry and has applications in various fields, including trigonometry, engineering, physics, and architecture.

The Pythagorean Theorem can be used in various ways, including:

  • Finding length of any one side of a right triangle when the lengths of the other two sides are known.
  • Checking if a Triangle is Right: You can use the theorem to check if a triangle is a right triangle by verifying if the relationship a² + b² = c² holds.
  • Solving Problems Involving Right Triangles: The theorem is used to solve problems involving right triangles, such as finding distances, heights, or angles in various real-world situations.

For example, if you have a right triangle with one side (a) measuring 3 units and the other side (b) measuring 4 units, you can use the Pythagorean Theorem to find the length of the hypotenuse (c):

c² = a² + b² c² = 3² + 4² c² = 9 + 16 c² = 25

To find c, take the square root of both sides:

c = √25 c = 5

So, the length of the hypotenuse (c) is 5 units. This illustrates how the Pythagorean Theorem can be used to solve for side lengths in a right triangle.

Click on the following worksheet to get a printable pdf document. Scroll down the page for more Pythagorean Theorem Worksheets .

Pythagorean Theorem Worksheet

More Pythagorean Theorem Worksheets

Printable (Answers on the second page.) Pythagorean Theorem Worksheet #1 (find hypotenuse) Pythagorean Theorem Worksheet #2 (find a side) Pythagorean Theorem Worksheet #3 (2-step) Pythagorean Theorem Worksheet #4 (check for right triangle) Pythagorean Theorem Worksheet #5 (word problems)

Online Pythagorean Theorem Find the missing side Test for right triangle Pythagorean Theorem Word Problems Converse Pythagorean Theorem, Types of Triangles

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MatterofMath

Pythagorean Theorem Worksheet: 4 Levels

Looking for the perfect Pythagorean theorem worksheet? Need some right-angle triangle practice? Look no further!

The theorem may be short and seem simple, but there are many types of questions you could face in an exam.

You will use this theorem if you continue studying math at a higher level. It’s a useful equation to keep in your math tool kit !

Applying the theorem with confidence is helpful for many people in the world.

The Pythagorean theorem really does have lots of real-life uses. You may use it for crafts, in construction, for baking, gardening, or choosing a smartphone. Without it, GPS location wouldn’t be a thing!

Each worksheet has a different focus. You might have one particular area you’d like to improve on, or perhaps you’re looking for a complete crash course.

Either way, don’t forget to show your working out, and start at the right skill level – each Pythagorean theorem worksheet is given in order of difficulty!

Worksheets The Man, the Legend: Pythagoras of Samos Who ACTUALLY Wrote the Pythagorean Theorem? What is the Theorem Other Theories and Discoveries Attributed to Pythagoras How Many Solutions? Solutions When a = 2 Special Solutions: Triples Properties of Triples A Brief Example Families of Triples A Briefer Example To Sum Up (Pun Intended!)

Free Printable Pythagorean Theorem Worksheet With Answers

Level 1: identify the right-angle triangles.

Pythagorean theorem worksheet 1 doc printable

Level 2: Identify the Pythagorean Triples

Pythagorean Theorem Worksheet 2 DOC printable

Level 3: Find the Missing Side Length of Right-Angle Triangles

Pythagorean Theorem Worksheet 3 DOC printable

Level 4: Real-World Pythagorean Theorem Word Problems (2 Pages)

Pythagorean Theorem Worksheet 4 DOC printable

The Man, the Legend: Pythagoras of Samos

Pythagoras sketch

The Pythagorean theorem is named after the Greek mathematician and philosopher, Pythagoras, who was born around 570 BC.

His home was the small Greek island Samos which is in the Mediterranean Sea, near Turkey.

Although he grew up in a beautiful paradise, there was political conflict. He traveled to Egypt and Persia as a young man before settling in southern Italy.

In Crotone, Italy, he established a school of philosophy and science. His students were called his disciples, his school like a family – a brotherhood.

In fact, the group he established was called The Brotherhood of Pythagoreans.

ROME, ITALY - MAY 07, 2019: School of Athens painting by Raphael, Vatican Museums, Vatican City

The Brotherhood had a structured life of exercise and study. They believed that to live forever in their second life , they needed to be physically perfect.

Pythagoras and his disciples, as they were called, were incredibly disciplined. They wanted to keep their bodies “morally pure”.

He was, by most accounts, a wacky guy – he always put his right shoe on first, never ate beans, didn’t travel by public roads, and encouraged his students to be silent wherever possible!

None of his writings have survived. That isn’t too surprising, given that he lived almost 2600 years ago!

Pythagoras is a man of mystery. Without his records, everything we know about him comes from what his disciples and other scholars wrote about him.

Imagine trying to find out what someone else is like, and what discoveries they have made, from their friends and family’s posts on social media. You wouldn’t get all the detail you need!

It’s known that he taught a special “Pythagorean Way of Life,” based on religion and ritual. Interestingly though, his beliefs were shaped by mathematics.

After he died in roughly 490BC, many people glorified Pythagoras and considered him God-like.

Apparently, he performed miracles: he killed a dangerous snake by biting it, he was seen in two cities at the same time, a river spoke to him and his thigh was golden.

Click here to go back to the Pythagorean theorem worksheet .

Who ACTUALLY Wrote the Pythagorean Theorem?

a^2+b^2=c^2 chalkboard

The Pythagorean Theorem is what Pythagoras is most famous for – it’s probably the first thing you think of when you hear his name.

But did Pythagoras himself really think of the theorem? Historians don’t know! It’s possible that one of his disciples wrote it after Pythagoras died, and named it after him.

The Pythagoreans were secretive. They came up with many proofs of important mathematical ideas but didn’t share their discoveries with anyone else.

If Pythagoras really did write the famous theorem, he didn’t tell the world about it!

What is the Pythagorean Theorem?

First, a right-angled triangle is a triangle that has an internal angle of 90°. It looks a bit like a ladder leaning between the floor and a wall.

Second, the hypotenuse of a right-angled triangle is its longest side. It is always the diagonal and never touches the right-angle. It would be the ladder in this picture!

Diagram ladder on wall

“In a right-angled triangle, the square of the hypotenuse equals the sum of squares of the other two sides”.

Don’t let the word ‘theorem’ put you off! It boils down to one equation:

Where a and b are the smaller side lengths and c is the length of hypotenuse .

If you have two lengths of a right-angled triangle, you can find the third using the formula.

Other Theories and Discoveries Attributed to Pythagoras

World Climate Zones

He was a well-rounded guy!

In Ancient Greek society, Pythagoras was credited with being the first to teach that the Earth is a sphere.

He was also one of the first to divide the world into five climatic zones – two cold zones at the poles, a tropical zone across the equator, and milder zones between them.

Pythagoras believed that math and abstract thought are the basis of morality and science. His teachings in philosophy inspired famous philosophers Plato and Aristotle.

Pythagoras started the idea of a number system, founded on his belief that numbers make up everything in the world.

He believed that all numbers are rational – that they can be written as fractions – and that they were beautiful in their simplicity.

The Pythagoreans were shocked when they discovered \(\sqrt{2\ }\), an irrational number that cannot be written as a fraction.

Pythagoras linked math to music when he heard blacksmiths’ hammers hitting metal. He realized that the pitch of the sound made was related to the size of the hammer.

Certain pitches sound nice when played together – they are in consonance. Other pitches clash, making a nasty sound – they are in dissonance.

Playing notes with special intervals between their pitches produces harmonies. Thank Pythagoras next time you hear singers harmonizing!

How Many Solutions?

Pythagoras theorem teaching cartoon

There are infinitely many solutions to the equation a 2 +b 2 =c 2

This becomes clear if one of the values is fixed. Let’s suppose that a=2 and see how many pairs of b and c values make sense.

This equation gives the relationship between b and c . Any value of c>2 has a corresponding value of b which satisfies the Pythagorean theorem because the radical is positive.

If c<2 , the equation has imaginary, or complex numbers .

For example, if we let c=1 , then

which isn’t a real number, because no real number can multiply by itself to make a negative number!

Solutions When a=2

We have just said that you can pick any value of c>2 and plug this into the equation \(b=\sqrt{c^2 – 4}\) to complete the solution.

Let’s give this a try:

If you stop to think about it, what would a right-angle triangle with two sides of the same length look like? Let’s use math to find out.

We’re going to let a=2 and c=2 , and put them into the equation to see what happens.

Hmmm, a right-angle triangle with one side of zero length? The math has spoken – there is no such thing!

There are infinitely many values of c to pick from, so there are infinitely many corresponding values of b , so there are infinitely many solutions to the Pythagorean Theorem when a=2 !

This thinking can be applied to any fixed value of a . There are infinitely many solutions for each fixed value of a , and there are infinitely many a ’s to pick from.

In short, that’s a lot of solutions!

Special Solutions: Pythagorean Triples

Lady holding out 3 fingers

The possibilities are literally endless when it comes to solving this equation.

It’s also interesting to study a subset of solutions, called Pythagorean triples .

These are triplets of positive integers a , b and c which solve the equation a 2 +b 2 =c 2 .

The first – and most famous! – example is 3 2 +4 2 =5 2 which, when used for measuring, is often referred to by tradespeople as 3, 4, 5.

Pythagorean triples are easier to work with because no radicals are involved. However, there is less freedom when creating them.

These triplets aren’t constructed by picking two variables and solving for the third – because of the special integer property, we must be more sophisticated!

Properties of Pythagorean Triples

The integer triplets \((a,b,c)\) fit into one of two categories: either all three numbers are even, or one is even and the other two are odd.

This is because the squares of even numbers are even, and the squares of odd numbers are odd.

Adding to this property, some equations help us construct a valid Pythagorean triple.

Begin by picking two positive integers. Call the bigger number x , and the smaller number y . It’s important that x and y are not equal to each other.

Substituting x and y into the following equations will give your triple:

Set x=10, and y=5. This is valid because \(x\neq y\) and \(x>y\).

The Pythagorean triple is

It’s written as:

In this case, two of the numbers are odd and one is even so it’s in the second category.

We can check that the triple satisfies the Pythagorean theorem by simply substituting in a , b , and c to find that:

Families of Triples

Family on 3 fingers

Equations can be manipulated by doing the same thing to both sides. Even though it has a fancy name, Pythagoras’ equation is no different!

If you multiply both sides of an equation by a positive integer, it’s still valid.

This idea is super useful – multiplying a triple by a positive integer will always give another triple!

Multiplying a triple is the same as multiplying both sides of the corresponding Pythagorean equation.

Each number in the triple is multiplied by a positive integer, giving another positive integer.

Think of it as scaling up the triangle, dragging it by the corner as you would with a photo or textbox on a computer.

Dividing triples is a dangerous game – if the divisor is not a factor of each number in the triple, the resulting numbers will not be integers so a Pythagorean triple will not be made. Disaster!

Every triple can be scaled upwards but not all can be scaled downwards.

When multiplying triples, you’ve got a lot of options. Infinitely many, in fact!

This means that there is an infinite family of triples, related to one another by multiplication.

In each ‘family’ of related triples, the smallest is given a special name.

A Pythagorean triple is called primitive if the greatest common divisor of a , b and c is 1.

Primitive triples are the simplest elements in their families – they cannot be divided to produce a smaller triple.

Let’s return to the previous triple we found: (75, 100, 125) .

This triple is not primitive because of the greatest common divisor, or GCD , which is 25, not 1.

Dividing each number by 25 gives another Pythagorean triple:

The division within the exponent looks a little suspicious, but we aren’t breaking any rules.

It is the same as dividing both sides by 25 2 , which is allowed!

To Sum Up (Pun Intended!)

I hope you’ve enjoyed learning some history along with your math. Pythagoras is a legendary figure, shrouded in mystery, but his theorem doesn’t have to be.

After using a Pythagorean theorem worksheet, you should have the confidence to identify right-angled triangles, find missing side lengths, identify Pythagorean triples, solve real-life problems, and use the theorem in trickier shapes.

Each worksheet uses the same amazing little equation a 2 +b 2 =c 2 , discovered around 2500 years ago.

Pythagoras is most famous for this theorem, but he had a busy life! One of his achievements was creating a circle of scholars called The Brotherhood of Pythagoreans.

They were influential in Ancient Greek society – Pythagoras’s work in philosophy inspired Plato and Aristotle, and the Brotherhood made many discoveries.

To count the number of solutions to his equation would take a long time – an infinite time, in fact! There is also an infinite number of integer solutions, called Pythagorean triples.

These triples are related to each other by multiplication . The first triple in each related ‘family’ is called primitive.

Once you’ve mastered using the theorem, you might enjoy learning about other mathematicians! The story of Fermat’s last is particularly interesting and begins with a similar equation…

If you have a question or comment to leave below, don’t be shy! It’s helpful to learn from each other.

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Mathematics LibreTexts

9.1: The Pythagorean Theorem

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  • Page ID 35449

  • David Arnold
  • College of the Redwoods

Pythagoras was a Greek mathematician and philosopher, born on the island of Samos (ca. 582 BC). He founded a number of schools, one in particular in a town in southern Italy called Crotone, whose members eventually became known as the Pythagoreans. The inner circle at the school, the Mathematikoi, lived at the school, rid themselves of all personal possessions, were vegetarians, and observed a strict vow of silence. They studied mathematics, philosophy, and music, and held the belief that numbers constitute the true nature of things, giving numbers a mystical or even spiritual quality.

Today, nothing is known of Pythagoras’s writings, perhaps due to the secrecy and silence of the Pythagorean society. However, one of the most famous theorems in all of mathematics does bear his name, the Pythagorean Theorem.

Prior to revealing the contents of the Pythagorean Theorem, we pause to provide the definition of a right triangle and its constituent parts.

Right Triangle

A triangle with one right angle (90 ◦ ) is called a right triangle . In the figure below, the right angle is marked with a little square.

imageedit_2_4590690964.png

The side of the triangle that is directly opposite the right angle is called the hypotenuse . The sides of the triangle that include the right angle are called the legs of the right triangle.

Now we can state one of the most ancient theorems of mathematics, the Pythagorean Theorem .

Pythagorean Theorem

Let \(c\) represent the length of the hypotenuse of a right triangle, and let a and b represent the lengths of its legs, as pictured in the image that follows.

imageedit_6_3162970942.png

The relationship involving the legs and hypotenuse of the right triangle, given by

\[a^2 + b^2 = c^2,\nonumber \]

is called the Pythagorean Theorem .

Here are two important observations.

Observations Regarding the Hypotenuse

Two important facts regarding the hypotenuse of the right triangle are:

  • The hypotenuse is the longest side of the triangle and lies directly opposite the right angle.
  • In the Pythagorean equation \(a^2 + b^2 = c^2\), the hypotenuse lies by itself on one side of the equation.

The Pythagorean Theorem can only be applied to right triangles.

Let’s look at a simple application of the Pythagorean Theorem.

The legs of a right triangle measure 3 and 4 meters, respectively. Find the length of the hypotenuse.

Let’s follow the Requirements for Word Problem Solutions .

1. Set up a Variable Dictionary . Let c represent the length of the hypotenuse, as pictured in the following sketch.

2. Set up an Equation . The Pythagorean Theorem says that

\[a^2 + b^2 = c^2.\nonumber \]

In this example, the legs are known. Substitute 4 for a and 3 for b (3 for a and 4 for b works equally well) into the Pythagorean equation.

\[4^2 + 3^2 = c^2\nonumber \]

3. Solve the Equation .

\[ \begin{aligned} 4^2 + 3^2 = c^2 ~ & \textcolor{red}{ \text{ The Pythagorean equation.}} \\ 16 + 9 = c^2 ~ & \textcolor{red}{ \text{ Exponents first: } 4^2 = 16 \text{ and } 3^2 = 9.} \\ 25 = c^2 ~ & \textcolor{red}{ \text{ Add: } 16+9=25.} \\ 5 = c~ & \textcolor{red}{ \text{ Take the nonnegative square root.}} \end{aligned}\nonumber \]

Technically, there are two answers to c 2 = 25, i.e., c = −5 or c = 5. However, c represents the hypotenuse of the right triangle and must be nonnegative. Hence, we must choose c = 5.

4. Answer the Question . The hypotenuse has length 5 meters.

5. Look Back . Do the numbers satisfy the Pythagorean Theorem? The sum of the squares of the legs should equal the square of the hypotenuse. Let’s check.

\[\begin{aligned} 4^2 + 3^2 = 5^2 \\ 16 + 9 = 25 \\ 25 = 25 \end{aligned}\nonumber \]

All is well!

The legs of a right triangle measure 5 and 12 feet, respectively. Find the length of the hypotenuse.

Given the following right triangle, find the length of the missing side.

imageedit_10_6811566794.png

Note that the hypotenuse (across from the right angle) has length 13. This quantity should lie on one side of the Pythagorean equation all by itself. The sum of the squares of the legs go on the other side. Hence,

\[5^2 + x^2 = 13^2\nonumber \]

Solve the equation for x .

\[ \begin{aligned} 25+x^2 = 169 ~ & \textcolor{red}{ \text{ Exponents first: } 5^2 = 25 \text{ and } 13^2 = 169.} \\ 25 + x^2 -25 = 169 - 25 ~ & \textcolor{red}{ \text{ Subtract 25 from both sides.}} \\ x^2 = 144 ~ & \textcolor{red}{ \text{ Simplify both sides.}} \\ x= 12 ~ & \textcolor{red}{ \text{ Take the nonnegative square root of 144.}} \end{aligned}\nonumber \]

The hypotenuse of a right triangle measures 25 centimeters. One leg of the right triangle measures 24 centimeters. Find the length of the remaining leg.

7 centimeters

Perfect squares are nice, but not required.

Given the following right triangle, find the exact length of the missing side.

imageedit_14_5088392393.png

Note that the hypotenuse (across from the right angle) has length 12. This quantity should lie on one side of the Pythagorean equation all by itself. The sum of the squares of the legs go on the other side. Hence,

\[x^2 + 7^2 = 12^2\nonumber \]

\[ \begin{aligned} x^2 + 49 = 144 ~ & \textcolor{red}{ \text{ Exponents first: } 7^2 = 49 \text{ and } 12^2 = 144.} \\ x^2 + 49-49 = 144 - 49 ~ & \textcolor{red}{ \text{ Subtract 49 from both sides.}} \\ x^2 = 95 ~ & \textcolor{red}{ \text{ Simplify both sides.}} \\ x = \sqrt{95} ~ & \textcolor{red}{ \text{ Take the nonnegative square root of 95.}} \end{aligned}\nonumber \]

Hence, the exact length of the missing side is \(\sqrt{95}\).

The hypotenuse and one leg of a right triangle measure 9 and 7 inches, respectively. Find the length of the remaining leg.

Add texts here. Do not delete this text first.

Important Observation

Any attempt to use your calculator to approximate 95 in Example 3 would be an error as the instructions asked for an exact answer.

Sometimes an approximate answer is desired, particularly in applications.

Ginny want to create a vegetable garden in the corner of her yard in the shape of a right triangle. She cuts two boards of length 8 feet which will form the legs of her garden. Find the length of board she should cut to form the hypotenuse of her garden, correct to the nearest tenth of a foot.

We follow the Requirements for Word Problem Solutions .

1. Set up a Variable Dictionary . We begin with a labeled sketch. Let x represent the length of the unknown hypotenuse.

imageedit_18_8045316766.png

2. Set Up an Equation . The hypotenuse is isolated on one side of the Pythagorean equation.

\[x^2 = 8^2 + 8^2\nonumber \]

\[ \begin{aligned} x^2 = 8^2 + 8^2 ~ & \textcolor{red}{ \text{ The Pythagorean equation.}} \\ x^2 = 64 + 64 ~ & \textcolor{red}{ \text{ Exponents first: } 8^2 = 64 \text{ and } 8^2 = 64.} \\ x^2 = 128 ~ & \textcolor{red}{ \text{ Add: } 64 + 64 = 128.} \\ x = \sqrt{128} ~ & \textcolor{red}{ \text{ Take the nonnegative square root.}} \end{aligned}\nonumber \]

4. Answer the Question . The exact length of the hypotenuse is \(\sqrt{128}\) feet, but we’re asked to find the hypotenuse to the nearest tenth of a foot. Using a calculator, we find an approximation for \(\sqrt{128}\).

\[\sqrt{128} \approx 11.313708499\nonumber \]

To round to the nearest tenth, first identify the rounding and test digits.

imageedit_22_8154063398.png

The test digit is less than five. So we leave the rounding digit alone and truncate. Therefore, correct to the nearest tenth of a foot, the length of the hypotenuse is approximately 11.3 feet.

5. Look Back . The sum of the squares of the legs is

\[ \begin{aligned} 8^2 + 8^2 = 64 + 64 \\ = 128. \end{aligned}\nonumber \]

The square of the hypotenuse is

\[(11.3)^2 = 127.69\nonumber \]

These are almost the same, the discrepancy due to the fact that we rounded to find an approximation for the hypotenuse.

A 15 foot ladder leans against the wall of a building. The base of the ladder lies 5 feet from the base of the wall. How high up the wall does the top of the ladder reach? Round your answer to the nearest tenth of a foot.

In Exercises 1-16, your solutions should include a well-labeled sketch.

1. The length of one leg of a right triangle is 15 meters, and the length of the hypotenuse is 25 meters. Find the exact length of the other leg.

2. The length of one leg of a right triangle is 7 meters, and the length of the hypotenuse is 25 meters. Find the exact length of the other leg.

3. The lengths of two legs of a right triangle are 12 meters and 16 meters. Find the exact length of the hypotenuse.

4. The lengths of two legs of a right triangle are 9 meters and 12 meters. Find the exact length of the hypotenuse.

5. The length of one leg of a right triangle is 13 meters, and the length of the hypotenuse is 22 meters. Find the exact length of the other leg.

6. The length of one leg of a right triangle is 6 meters, and the length of the hypotenuse is 15 meters. Find the exact length of the other leg.

7. The lengths of two legs of a right triangle are 2 meters and 21 meters. Find the exact length of the hypotenuse.

8. The lengths of two legs of a right triangle are 7 meters and 8 meters. Find the exact length of the hypotenuse.

9. The length of one leg of a right triangle is 12 meters, and the length of the hypotenuse is 19 meters. Find the exact length of the other leg.

10. The length of one leg of a right triangle is 5 meters, and the length of the hypotenuse is 10 meters. Find the exact length of the other leg.

11. The lengths of two legs of a right triangle are 6 meters and 8 meters. Find the exact length of the hypotenuse.

12. The lengths of two legs of a right triangle are 5 meters and 12 meters. Find the exact length of the hypotenuse.

13. The length of one leg of a right triangle is 6 meters, and the length of the hypotenuse is 10 meters. Find the exact length of the other leg.

14. The length of one leg of a right triangle is 9 meters, and the length of the hypotenuse is 15 meters. Find the exact length of the other leg.

15. The lengths of two legs of a right triangle are 6 meters and 22 meters. Find the exact length of the hypotenuse.

16. The lengths of two legs of a right triangle are 9 meters and 19 meters. Find the exact length of the hypotenuse.

In Exercises 17-24, your solutions should include a well-labeled sketch.

17. The lengths of two legs of a right triangle are 3 meters and 18 meters. Find the length of the hypotenuse. Round your answer to the nearest hundredth.

18. The lengths of two legs of a right triangle are 10 feet and 16 feet. Find the length of the hypotenuse. Round your answer to the nearest tenth.

19. The length of one leg of a right triangle is 2 meters, and the length of the hypotenuse is 17 meters. Find the length of the other leg. Round your answer to the nearest tenth.

20. The length of one leg of a right triangle is 4 meters, and the length of the hypotenuse is 12 meters. Find the length of the other leg. Round your answer to the nearest hundredth.

21. The lengths of two legs of a right triangle are 15 feet and 18 feet. Find the length of the hypotenuse. Round your answer to the nearest hundredth.

22. The lengths of two legs of a right triangle are 6 feet and 13 feet. Find the length of the hypotenuse. Round your answer to the nearest tenth.

23. The length of one leg of a right triangle is 4 meters, and the length of the hypotenuse is 8 meters. Find the length of the other leg. Round your answer to the nearest hundredth.

24. The length of one leg of a right triangle is 3 meters, and the length of the hypotenuse is 15 meters. Find the length of the other leg. Round your answer to the nearest tenth.

25. Greta and Fritz are planting a 13-meter by 18-meter rectangular garden, and are laying it out using string. They would like to know the length of a diagonal to make sure that right angles are formed. Find the length of a diagonal. Round your answer to the nearest hundredth. Your solution should include a well-labeled sketch.

26. Markos and Angelina are planting an 11- meter by 19-meter rectangular garden, and are laying it out using string. They would like to know the length of a diagonal to make sure that right angles are formed. Find the length of a diagonal. Round your answer to the nearest tenth. Your solution should include a well-labeled sketch.

27. The base of a 24-meter long guy wire is located 10 meters from the base of the telephone pole that it is anchoring. How high up the pole does the guy wire reach? Round your answer to the nearest hundredth. Your solution should include a well-labeled sketch.

28. The base of a 30-foot long guy wire is located 9 feet from the base of the telephone pole that it is anchoring. How high up the pole does the guy wire reach? Round your answer to the nearest hundredth. Your solution should include a well-labeled sketch.

29. Hiking Trail. A hiking trail runs due south for 8 kilometers, then turns west for about 15 kilometers, and then heads northeast on a direct path to the starting point. How long is the entire trail?

30. Animal Trail. An animal trail runs due east from a watering hole for 12 kilometers, then goes north for 5 kilometers. Then the trail turns southwest on a direct path back to the watering hole. How long is the entire trail?

31. Upper Window. A 10-foot ladder leans against the wall of a house. How close to the wall must the bottom of the ladder be in order to reach a window 8 feet above the ground?

32. How high? A 10-foot ladder leans against the wall of a house. How high will the ladder be if the bottom of the ladder is 4 feet from the wall? Round your answer to the nearest tenth.

1. 20 meters

3. 20 meters

5. √315 meters

7. \(\sqrt{445}\) meters

9. \(\sqrt{217}\) meters

11. 10 meters

13. 8 meters

15. \(\sqrt{520}\) meters

17. 18.25 meters

19. 16.9 meters

21. 23.43 feet

23. 6.93 meters

25. 22.20 meters

27. 21.82 meters

29. 40 kilometers

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  1. 48 Pythagorean Theorem Worksheet with Answers [Word + PDF]

    pythagorean theorem worksheet answers sheet 1

  2. Pythagorean Theorem Worksheet With Answers

    pythagorean theorem worksheet answers sheet 1

  3. Printable Pythagorean Theorem Worksheet

    pythagorean theorem worksheet answers sheet 1

  4. Pythagorean Theorem Practice with Answer Key by Making It Teacher

    pythagorean theorem worksheet answers sheet 1

  5. Pythagoras Theorem Worksheet With Answers

    pythagorean theorem worksheet answers sheet 1

  6. Pythagorean Theorem Worksheet Answer Key

    pythagorean theorem worksheet answers sheet 1

VIDEO

  1. 8.1 Apply the Pythagorean Theorem

  2. pythagorean theorem in 8 SECONDS?!

  3. THE PYTHAGOREAN THEOREM#shorts#math

  4. Math Performance Task: Pythagorean Theorem

  5. Pythagorean Theorem Worksheet #1

  6. 9.2 Pythagorean Theorem

COMMENTS

  1. 48 Pythagorean Theorem Worksheet with Answers [Word + PDF]

    Formulated in the 6th Century BC by Greek Philosopher and mathematician Pythagoras of Samos, Pythagorean Theorem is a mathematic equation used for a variety of purposes. Over the years, many engineers and architects have used Pythagorean Theorem worksheet to complete their projects. Table of Contents [ Show]

  2. PDF Finding an Unknown Side

    Use the Pythagorean Theorem to test the triangles shown or described in each problem below. Is it a . right. triangle? 2 4 6 1 3 5. 5 5. 4 4. 4 41. 3 13. 32 If a triangle has sides that are 12, 10 and 6 . meters long, is it a right triangle? Is a triangle with side lengths of 4, 5, and . 6 inches a right triangle?

  3. Pythagorean Theorem Worksheets

    Unknown Side of a Right Triangle Apply Pythagorean theorem to find the unknown side of the right triangle. Round the answer to the nearest tenth. Download the set Unknown Length: Shapes Apply the Pythagorean theorem to find the unknown length of each shape in these printable worksheets. Round the answer to the nearest tenth. Download the set

  4. Pythagorean Theorem Worksheets (printable, online, answers, examples)

    There are five sets of Pythagorean Theorem worksheets: Find the length of the hypotenuse Find the length of a side Find the length in two steps Check for right triangle Pythagorean Theorem Word Problems The Pythagorean Theorem also called the Pythagoras' Theorem is a fundamental principle in geometry that relates the sides of a right triangle.

  5. Pythagorean Theorem Worksheet: 4 Levels · Matter of Math

    To Sum Up (Pun Intended!) Free Printable Pythagorean Theorem Worksheet With Answers Level 1: Identify the Right-Angle Triangles Level 2: Identify the Pythagorean Triples Level 3: Find the Missing Side Length of Right-Angle Triangles Level 4: Real-World Pythagorean Theorem Word Problems (2 Pages) The Man, the Legend: Pythagoras of Samos

  6. Pythagorean Theorem Worksheets

    This worksheet explains how to use the Pythagorean theorem. A sample problem is solved, and two practice problems are provided. Introduction Worksheet Students will use the theorem to solve these triangle based problems. Practice Sheet Example problem: Let the distance of the ship from its starting point be x km.

  7. PDF The Pythagorean Theorem Date Period

    The Pythagorean Theorem Date_____ Period____ Do the following lengths form a right triangle? 1) 6 8 9 No 2) 5 12 13 Yes 3) 6 8 10 Yes 4) 3 4 5 Yes ... Create your own worksheets like this one with Infinite Pre-Algebra. Free trial available at KutaSoftware.com. Title: Pythagorean Theorem

  8. 20 Pythagorean Theorem Worksheet with Answers (Word & PDF)

    A Pythagorean Theorem Worksheet is a workbook containing different triangle orientations requiring a student to fill in the missing values using the Pythagorean Theorem. It is a great study tool for geometry students. Pythagorean Theorem Worksheet Templates & Examples #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 #11 #12 #13 #14 #15 #16 #17 #18 #19 #20

  9. Pythagorean Theorem Problems Worksheets

    This Pythagorean Theorem Problems Worksheet will produce problems for practicing solving the lengths of right triangles. You may choose the type of numbers and the sides of the triangle. This worksheet is a great resources for the 6th Grade, 7th Grade, and 8th Grade.

  10. PDF 1 Pythagoras' Theorem

    1.Verify this formula holds for the cases n= 1,2and 3above. P(1) : 1= 12 2 = 1 P(2) : 1+2= 23 2 P(3) : 1+2+3= 34 2 Does this formula hold for n= 100? How can we be sure? The answer is mathematical proof, of course! We need to prove that this formula holds for all n. We are going to start the induction by proving that P(4) holds without using ...

  11. PDF Worksheet #1

    Worksheet #1 - Word Problems Pythagorean Theorem Directions: Solve by drawing a picture, identifying a, b, and c, and applying the Pythagorean Theorem. Do not forget to give your answer with units and show ALL your work to receive full credit. 1. Two sides of a right triangle are 8 inches and 12 inches. a.

  12. PDF Pythagorean Theorem Sheet 1

    Answer Key Pythagorean Theorem Sheet 1 Score : Printable Math Worksheets @ www.mathworksheets4kids.com Name : 4) XZ = 5) EF = 6) JL = 14 yd? 3 yd Z Y X t? 8 ft F G E 17 m? 13 m K L J ... Answer Key Score : Printable Math Worksheets @ www.mathworksheets4kids.com Name : Sheet 3 4) TU = 5) EG = 6) QR = 8 ft? 6 ft S U T 3 m? 17 m E F G 1 yd? d Q R P

  13. Pythagorean Theorem Practice Problems With Answers

    Here are eight (8) Pythagorean Theorem problems for you to solve. You might need to find either the leg or the hypotenuse of the right triangle. These problems vary in type and difficulty, providing you an opportunity to level up your skills.

  14. PDF Missing Sides

    Title: missing-sides-1.pdf Author: USER 1 Created Date: 12/16/2023 2:47:57 PM

  15. PDF PYTHAGOREAN THEOREM

    PYTHAGOREAN THEOREM - WORKSHEET. For each triangle find the missing length. Round your answer to the nearest tenth. Then find the area and the perimeter. 10. Find a third number so that the three numbers form a right triangle: 11. Ms. Green tells you that a right triangle has a hypotenuse of 13 and a leg of 5.

  16. PDF Pythagorean Theorem Worksheets

    Solve for x 12 12 x x 8 16 16 x 30 15 x 10 30 x 8 4 x Name: Date: Score: /10 #1 #2 #3 #4. Title: Pythagorean Theorem Worksheets Created Date: 1/27/2022 11:49:10 AM

  17. PDF Pythagorean Theorem

    1 gure. C B ? N 6 yd L XY = H E EF = 15 mm ? G F 6) LM = ? 3 in W SW = V U V S

  18. PDF The Pythagorean Theorem and Its Converse

    Add. The sum of the squares of the two shorter sides equals the square of the longest side, so the. triangle is a right triangle. Exercises. Determine whether each set of measures can be the measures of the sides of a. right triangle. Then state whether they form a Pythagorean triple. 1. 30, 40, 50.

  19. 9.1: The Pythagorean Theorem

    The Pythagorean Theorem says that. a2 +b2 = c2. a 2 + b 2 = c 2. In this example, the legs are known. Substitute 4 for a and 3 for b (3 for a and 4 for b works equally well) into the Pythagorean equation. 42 +32 = c2 4 2 + 3 2 = c 2. 3. Solve the Equation. 42 +32 = c2 16 + 9 = c2 25 = c2 5 = c The Pythagorean equation.

  20. PDF Pythagorean theorem word problems worksheets with answers 1

    1. A swimmer starts at a point on the shore and swims 200 meters due east. Then, the swimmer turns and swims 300 meters due north. What is the shortest distance for the swimmer to return to the starting point to the nearest metre? a 2 + b 2 = c. 2. 2 200 + 300 = c. 2 2 Therefore the shortest distance. c 300 m is approximately 114 m.

  21. PDF Pythagorean Triples Sheet 1

    Pythagorean Triples Sheet 1 Printable Math Worksheets @ www.mathworksheets4kids.com Name : Score : 1) 24, 18, 30 12, 20, 16 29, 20, 21 3, ,14 10 ... A B2 cm C 6, 10, 8 8, 15, 17 5, 11, 9 1, 7, 5. Determine whether each set of numbers form a Pythagorean triple. Answer Key Pythagorean Triples Sheet 1 Printable Math Worksheets @ www ...

  22. Pythagoras Practice Questions

    The Corbettmaths Practice Questions on Pythagoras. Next: Direct and Inverse Proportion Practice Questions

  23. PDF Pythagorean Theorem

    8 ft Rachel bought a rug for her apartment. The rug is 11 feet long and 9 feet wide. Find the diagonal length of the rug. ? 11 ft 1) Mark is on his way home from work. He drives 35 miles due North and then 42 miles due West. Find the shortest distance he can cover to reach home early. N 42 mi ? 54.7 mi W E S 2) Joshua won a laptop in a school ra!e.