Free Printable converse pythagoras theorem worksheets

Discover a collection of free printable math worksheets focusing on the converse Pythagoras theorem, designed to help students and teachers explore this fundamental concept in geometry. Dive into these resources and enhance your math lessons today!

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Converse Pythagoras theorem worksheets are an essential resource for teachers who want to help their students excel in Math and Geometry. These worksheets provide a variety of problems and exercises that focus on the converse of the Pythagorean theorem, allowing students to practice and develop their understanding of this fundamental concept. By incorporating these worksheets into their lesson plans, teachers can ensure that their students are well-prepared for more advanced topics in Geometry. Additionally, these worksheets cater to different learning styles and can be easily adapted to suit the needs of each individual student. With a strong foundation in the converse Pythagoras theorem, students will be better equipped to tackle more complex mathematical problems and succeed in their academic pursuits.

Quizizz is an excellent platform that offers a wide range of educational resources, including converse Pythagoras theorem worksheets, to support teachers in their efforts to provide engaging and effective Math and Geometry lessons. By utilizing Quizizz, teachers can access a vast library of worksheets, quizzes, and other interactive learning materials that are designed to help students grasp important mathematical concepts. Furthermore, Quizizz allows teachers to track their students' progress and identify areas where additional support may be needed. This invaluable tool enables educators to create customized learning experiences that cater to the unique needs of each student, ensuring that they are well-prepared for success in Math and Geometry. By incorporating Quizizz into their teaching strategies, teachers can enhance their students' understanding of the converse Pythagoras theorem and other essential mathematical concepts.

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Pythagorean Theorem Worksheets

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Get into high gear with our free, printable Pythagorean theorem worksheets! Pythagoras's theorem plays a role in topics like trigonometry. Our Pythagorean theorem worksheet pdfs include finding the hypotenuse, identifying Pythagorean triples, identifying a right triangle using the converse of the theorem, and more!

Our Pythagorean theorem worksheets work best for 7th grade, 8th grade, and high school students.

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Area of Right Triangles

Memorize the relation between the hypotenuse and legs of a right triangle (c 2 = a 2 + b 2 ) and figure out what Pythagorean triplets are! This geometrical interpretation of the Pythagorean theorem is buoyed by areas of squares.

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A Pythagorean triple is a set of 3 positive integers that satisfy the Pythagorean theorem. A triple consists of three even numbers or two odd numbers and an even number. Use this fact to check if the numbers form a Pythagorean triple.

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Does the equation c 2 = a 2 + b 2 help find the length of a line segment? Recognize the distance formula as a variant of the Pythagorean theorem and measure the line segments on xy-planes in these pdf worksheets for grade 8 and high school!

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Converse of the Pythagorean Theorem | Is It a Right Triangle?

A triangle's dimensions are 10 units, 24 units, and 26 units. Is it a right triangle? Verify using the Pythagorean triples or the theorem's converse. These Pythagorean theorem worksheets have more!

Converse of the Pythagorean Theorem - Is It a Right Triangle?

The Pythagorean theorem helps find the length of the hypotenuse using the leg lengths. Add the squares of the two lengths and take the square root of the sum to arrive at the length of the hypotenuse.

Finding the Hypotenuse

Grade 7, grade 8, and high school kids draw on our printable Pythagorean theorem worksheets, presenting right triangles with a side length missing. Use the equation c 2 = a 2 + b 2 and solve.

Finding the Unknown Side Lengths in Right Triangles

Explore real-world situations that require the application of the Pythagorean theorem and solving for the missing length of the right triangle in this bunch of worksheets!

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Do you see two right triangles on a rectangle? Discern the diagonal of a rectangle as the hypotenuse of a right triangle, thereby applying the Pythagorean formula to work out the diagonal length.

Finding the Length of the Diagonal of a Rectangle

Explore the area of right triangles with these printable worksheets. Substitute the hypotenuse and known leg in the Pythagorean theorem to find the other leg and figure out the area of the right triangle.

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CONVERSE OF THE PYTHAGOREAN THEOREM WORKSHEET

Problem 1 :

Determine whether triangle with the side lengths given below is a right triangle.

9 inches, 40 inches, and 41 inches

Problem 2 :

8 meters, 10 meters, and 12 meters

Problem 3 :

14 cm, 23 cm, and 25 cm

Problem 4 :

27 ft, 36 ft and 45 ft

converse of the pythagorean theorem worksheet pdf

1. Answer :

Let a = 9, b = 40, and c = 41.

(Always assume the longest side as "c")

Find the value of (a 2  + b 2 ). 

a 2  + b 2   =  9 2  + 40 2

a 2  + b 2   =  81 + 1600

a 2  + b 2   =  1681 ----- (1)

Find the value of c 2 . 

c 2   =  41 2

c 2   =  1681 ----- (2)

From (1) and (2), we get

a 2  + b 2   =  c 2

By the converse of Pythagorean theorem, the triangle with the side lengths  9 inches, 40 inches, and 41 inches is a right triangle. 

2. Answer :

Let a = 8, b = 10, and c = 12.

a 2  + b 2   =  8 2  + 10 2

a 2  + b 2   =  64 + 100

a 2  + b 2   =  164 ----- (1)

Find the value of  c 2 . 

c 2   =  12 2

c 2   =  144 ----- (2)

a 2  + b 2    ≠   c 2

By the converse of Pythagorean theorem, the triangle with the side lengths  8 meters, 10 meters, and 12 meters  is not a right triangle. 

3. Answer :

Let a = 14, b = 23, and c = 25.

a 2  + b 2   =  14 2  + 23 2

a 2  + b 2   =  196 + 529

a 2  + b 2   =  725 ----- (1)

Find the value of c ² . 

c 2   =  25 2

c 2   =  625 ----- (2)

By the converse of Pythagorean theorem, the triangle with the side lengths  14 cm, 23 cm, and 25 cm  is not a right triangle. 

4. Answer :

Let a = 27, b = 36, and c = 45.

a 2  + b 2   =  27 2  + 36 2

a 2  + b 2   =  729 + 1296

a 2  + b 2   =  2025 -----(1)

c 2   =  45 2

c 2   =  2025 -----(2)

a 2  + b 2   =  c 2

By the converse of Pythagorean theorem, the triangle with the side lengths  27 ft, 36 ft and 45 ft  is a right triangle.

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  • Converse Of Pythagoras Theorem

Converse of Pythagoras Theorem

The converse of Pythagoras theorem  states that “If the square of a side is equal to the sum of the square of the other two sides, then triangle must be right angle triangle”. Whereas Pythagorean theorem states that the sum of the square of two sides (legs) is equal to the square of the hypotenuse of a right-angle triangle. But, in the reverse of the Pythagorean theorem, it is said that if this relation satisfies, then triangle must be right angle triangle. So, if the sides of a triangle have length, a, b and c and satisfy given condition a 2 + b 2 = c 2 , then the triangle is a right-angle triangle.

Let us see the proof of this theorem along with examples.

Converse of Pythagoras Theorem Proof

Statement: If the length of a triangle is a, b and c and c 2 = a 2 + b 2 , then the triangle is a right-angle triangle.

Converse of Pythagoras theorem

Proof: Construct another triangle, △EGF, such as AC = EG = b and BC = FG = a.

Converse of Pythagorean Theorem Proof

In △EGF, by Pythagoras Theorem:

EF 2 = EG 2 + FG 2 = b 2 + a 2 …………(1)

In △ABC, by Pythagoras Theorem:

AB 2 = AC 2 + BC 2 = b 2 + a 2 …………(2)

From equation (1) and (2), we have;

EF 2 = AB 2

⇒ △ ACB ≅ △EGF (By SSS postulate)

⇒ ∠G is right angle

Thus, △EGF is a right triangle.

Hence, we can say that the converse of Pythagorean theorem also holds.

Hence Proved.

As per the converse of the Pythagorean theorem, the formula for a right-angled triangle is given by:

Where a, b and c are the sides of a triangle.

Applications

Basically, the converse of the Pythagoras theorem is used to find whether the measurements of a given triangle belong to the right triangle or not. If we come to know that the given sides belong to a right-angled triangle, it helps in the construction of such a triangle. Using the concept of the converse of Pythagoras theorem, one can determine if the given three sides form a Pythagorean triplet.

Converse of Pythagoras Theorem Examples

Question 1: The sides of a triangle are 5, 12 and 13. Check whether the given triangle is a right triangle or not?

Solution: Given,

By using the converse of Pythagorean Theorem,

a 2 +b 2 = c 2

c 2 = a 2 +b 2

Substitute the given values in the above equation,

13 2 = 5 2 +12 2

169 = 25 + 144

So, the given lengths are does not satisfy the above condition.

Therefore, the given triangle is a right triangle.

Question 2: The sides of a triangle are 7, 11 and 13. Check whether the given triangle is a right triangle or not?

Solution: Given;

Substitute the given values in the the above equation,

13 2 = 7 2 + 11 2

169 = 49 + 121

So, it is not satisfied with the above condition.

Therefore, the given triangle is not a right triangle.

Question 3: The sides of a triangle are 4,6 and 8. Say whether the given triangle is a right triangle or not.

Solution: Given: a = 4, b = 6, c = 8

By the converse of Pythagoras theorem

8 2 = 4 2 + 6 2

64 = 16 + 36

The sides of the given triangle do not satisfy the condition a 2 +b 2 = c 2 .

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Converse of Pythagoras Theorem

The converse of Pythagoras theorem is the reverse of the Pythagoras theorem and it helps in determining if a triangle is acute, right, or obtuse if the sum of the squares of two sides of a triangle is compared to the square of its third side. The Pythagorean theorem is the most used in trigonometry. Let us learn more about the converse of the Pythagoras theorem, the proof, and solve a few examples.

What is the Converse of Pythagoras Theorem?

The converse of Pythagoras theorem states that if the square of the length of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle. The converse is the complete reverse of the Pythagoras theorem. The main application of the converse of the Pythagorean theorem is that the measurements help in determining the type of triangle - right, acute, or obtuse. Once the triangle is identified, constructing that triangle becomes simple. There are three cases that occur:

1. If the sum of the squares of two sides of a triangle is considered equivalent to the square of the hypotenuse, the triangle is a right triangle .

2. If the sum of the squares of two sides of a triangle is less than the square of the hypotenuse, the triangle is an obtuse triangle .

3. If the sum of the squares of two sides of a triangle is greater than the square of the hypotenuse, the triangle is an acute triangle .

Pythagoras Theorem

The Pythagoras theorem states that if a triangle is right-angled (90 degrees), then the square of the hypotenuse is equal to the sum of the squares of the other two sides. In the given triangle ABC, we have BC 2 = AB 2 + AC 2 ​​. Here, ​​​​AB is the base, AC is the altitude or the height, and BC is the hypotenuse. In other words, we can say, in a right triangle, (Opposite) 2 + (Adjacent) 2 = (Hypotenuse) 2 ​​​​​​.

Converse of Pythagoras Theorem

Proof of Converse of Pythagoras Theorem

Statement: In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.

Proof: Here, we are given a triangle ABC in which AC 2 = AB 2 + BC 2 . We need to prove that ∠B = 90°.

To start with, we construct a ΔPQR right-angled at Q such that PQ = AB and QR = BC.

Converse of Pythagoras Theorem

Now, from Δ PQR, we have:

PR 2 = PQ 2 + QR 2 (Pythagoras Theorem, as ∠Q=90°)

or PR 2 = AB 2 + BC 2 (By construction)........ (1)

But AC 2 = AB 2 + BC 2 (Given).......... (2)

So, AC = PR (From (1) and (2)).............. (3)

Now, in ΔABC and ΔPQR,

AB = PQ (By construction)

BC = QR (By construction)

AC = PR (Proved in (3))

So, ΔABC ≃ ΔPQR (According to the SSS congruence)

∠B = ∠Q (Corresponding angles of congruent triangles)

∠Q = 90° (By construction)

So ∠B = 90°.

Hence, the converse of the Pythagoras theorem is proved.

Converse of Pythagoras Theorem Formula

The converse of Pythagoras theorem formula is c 2 = a 2 + b 2 , where a, b, and c are the sides of the triangle.

Related Topics

Listed below are a few topics related to the converse of the Pythagoras theorem, take a look.

  • Right Triangle Formulas
  • Hypotenuse Leg Theorem
  • What is Similarity
  • Similarity in Triangles

Examples on Converse of Pythagoras Theorem

Example 1: The side of the triangle is of length 8 units, 10 units, and 6 units. Is this triangle a right triangle? If so, which side is the hypotenuse?

We know that the hypotenuse is the longest side in a triangle. The side or lengths is given as 8 units, 10 units, and 6 units. Therefore, 10 units is the hypotenuse.

Using the converse of Pythagoras theorem, we get,

(10) 2 = (8) 2 + (6) 2

100 = 64 + 36

Since both sides are equal, the triangle is a right triangle.

Example 2: Check if the triangle is acute, right, or an obtuse triangle with side lengths as 6, 8, and 11 units.

Solution: According to the length, we know that 11 units are the longest side.

Compare the square lengths of both the sides in the equation c 2 = a 2 + b 2 .

(11) 2 = (6) 2 + (8) 2

121 = 36 + 64

Hence, (11) 2 > (6) 2 + (8) 2

Therefore, according to the application of converse of Pythagoras theorem (If the sum of the squares of two sides of a triangle is less than the square of the hypotenuse, the triangle is an obtuse triangle), the triangle is an obtuse triangle.

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Practice Questions on Converse of Pythagoras Theorem

Faqs on converse of pythagoras theorem.

The coverse of the Pythagoras theore m states that, in a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.

What is the Formula for Converse of Pythagoras Theorem?

The converse of Pythagoras theorem formula is c 2 = a 2 + b 2 , where a, b, and c are the sides of the triangle .

How Do You Prove the Converse of Pythagoras Theorem?

The converse of the Pythagoras theorem states that if the square of the length of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle. If we consider two triangles ΔABC and ΔPQR where c 2 = a 2 + b 2 then we can say ∠C is a right triangle.

What is the Converse of the Pythagoras Theorem Useful For?

The converse of the Pythagoras theorem is useful in determining if a triangle is a right triangle or not. Whereas, a Pythagorean theorem helps in determining the length of the missing side of a right triangle.

What is the Application of the Converse of Pythagoras Theorem?

The application of the converse of the Pythagoras theorem is that the measurements help in determining what type of a triangle it is. There are three scenarios that we can determine, they are:

  • If the sum of the squares of two sides of a triangle is considered equivalent to the square of the hypotenuse, the triangle is a right triangle.
  • If the sum of the squares of two sides of a triangle is less than the square of the hypotenuse, the triangle is an obtuse triangle.
  • If the sum of the squares of two sides of a triangle is greater than the square of the hypotenuse, the triangle is an acute triangle.

What is the Pythagorean Theorem?

The Pythagoras theorem states that if a triangle is right-angled (90 degrees), then the square of the hypotenuse is equal to the sum of the squares of the other two sides. We can say, (Opposite) 2 + (Adjacent) 2 = (Hypotenuse) 2 ​​​​​​.

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  4. PDF The Pythagorean Theorem 8-1 and Its Converse

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  5. PDF The Pythagorean Theorem and Its Converse Worksheet

    Name Remember The Converse of the Pythagorean Theorem The Pythagorean Theorem can be used to determine whether a triangle is right, acute, or obtuse. Think of the long side as c and the two shorter sides as a and b.

  6. PDF Geometry Worksheet 7.2

    Geometry Worksheet 7.2 - Converse of the Pythagorean Theorem 1. Write the Pythagorean Theorem Name ________________ Extension of the Pythagorean Theorem 2. If the triangle is acute then c2 ____ a2 + b2 3. If the triangle is obtuse then c2 _____ a2 + b2 Are the triangles below right triangles? If not, what kind of triangles are they? 4. 9 15

  7. PDF Lesson 16: The Converse of the Pythagorean Theorem

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  9. Converse of the Pythagorean Theorem Worksheets

    Give children access to our pdf converse of the Pythagorean theorem worksheets, so they develop proficiency in finding whether a triangle, with the three side dimensions given, is a right triangle or not. Guide the enthusiastic minds to apply the Pythagorean theorem and determine whether the given triangle is a right triangle.

  10. PDF Pythagorean Theorem and Its Converse

    The Pythagorean Theorem and Its Converse. Find the missing side of each triangle. Round your answers to the nearest tenth if necessary. Find the missing side of each triangle. Leave your answers in simplest radical form. Find the missing side of each right triangle. Side c is the hypotenuse.

  11. PDF Notes 5-7: Pythagorean Theorem

    Converse of the Pythagorean Theorem: If the square of one side of a triangle is equal to the ______ of the ______________ of the other two sides, then the triangle is a _________ triangle. Decide if the following lengths can form a right triangle. 14) 6, 8, 10 ______ 15) 10, 24, 26 ______ 16) 9, 12, 18 ______ 17) 16, 30, 34 ______

  12. 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

  13. PDF The Pythagorean Theorem 9.2 Geometry

    The Converse of the Pythagorean Theorem states that: If the lengths of the sides of a triangle satisfy the Pythagorean Theorm, then the triangle is a right triangle. Examples: Determine which of the following is a right triangle? 21 20 16 29 65 63 17 8 15 Finding all pythagorean triples:

  14. PDF Pythagoras pdf

    Question 2: Shown is a square with side length 5cm. Find the length of the diagonal, x. Question 3: Shown is a right angle triangle. Calculate: the perimeter of the triangle. the area of the triangle. Question 4: A rectangle is 20cm long and 8cm wide. Find the length of the diagonal of the rectangle. Question 5: An airplane is Ylying from ...

  15. Free Printable converse pythagoras theorem worksheets

    Discover a collection of free printable math worksheets focusing on the converse Pythagoras theorem, designed to help students and teachers explore this fundamental concept in geometry. Dive into these resources and enhance your math lessons today! converse pythagoras theorem Pythagorean Theorem Vocabulary 10 Q 8th - 12th

  16. Converse Of Pythagorean Theorem Teaching Resources

    PDF. This ready to use product is a quick, fun way to have your students practice the converse of the Pythagorean Theorem. Students will be shown 15 different triangles and will need to determine if they are right triangles using the given side lengths. Instead of writing their answers, students will color them!

  17. Pythagorean Theorem Worksheets

    Pythagoras's theorem plays a role in topics like trigonometry. Our Pythagorean theorem worksheet pdfs include finding the hypotenuse, identifying Pythagorean triples, identifying a right triangle using the converse of the theorem, and more! Our Pythagorean theorem worksheets work best for 7th grade, 8th grade, and high school students.

  18. Converse of The Pythagorean Theorem Worksheet

    By the converse of Pythagorean theorem, the triangle with the side lengths 9 inches, 40 inches, and 41 inches is a right triangle. 2. Answer : 8 meters, 10 meters, and 12 meters. Step 1 : Let a = 8, b = 10, and c = 12. (Always assume the longest side as "c") Step 2 : Find the value of (a2 + b2).

  19. 48 Pythagorean Theorem Worksheet with Answers [Word + PDF]

    A simple equation, Pythagorean Theorem states that the square of the hypotenuse (the side opposite to the right angle triangle) is equal to the sum of the other two sides. Following is how the Pythagorean equation is written: a²+b²=c². In the aforementioned equation, c is the length of the hypotenuse while the length of the other two sides ...

  20. Converse of Pythagoras Theorem

    The converse of Pythagoras theorem states that "If the square of a side is equal to the sum of the square of the other two sides, then triangle must be right angle triangle". Whereas Pythagorean theorem states that the sum of the square of two sides (legs) is equal to the square of the hypotenuse of a right-angle triangle.

  21. PDF Name Date The Pythagorean Packet

    1. You can only use the Pythagorean Theorem on a RIGHT triangle (one with a 90° angle). 2. For any triangle, if a2 + b2 = c2 holds true, then that triangle is a RIGHT triangle. 3. It doesn't really matter what leg (side) you label a or b, what matters is that c is the HYPOTENUSE (located directly opposite the 90° angle.)

  22. Converse of Pythagoras Theorem

    Converse of Pythagoras Theorem. The converse of Pythagoras theorem is the reverse of the Pythagoras theorem and it helps in determining if a triangle is acute, right, or obtuse if the sum of the squares of two sides of a triangle is compared to the square of its third side. The Pythagorean theorem is the most used in trigonometry.

  23. 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.