converse to the pythagorean theorem assignment quizlet

The Converse of the Pythagorean Theorem

New york state common core math grade 8, module 7, lesson 16.

Lesson 16 Student Outcomes

Lesson 16 Summary

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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 ​​​​​​.

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.

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 ​​​​​​.

• Math Article
• 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.

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

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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Pythagorean Theorem and its Converse

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• 10. Multiple Choice Edit 15 minutes 1 pt You need a ladder that will reach up a 25 foot tall house when placed 10 feet away from the house. How long does the ladder need to be? 25 feet 26 feet 27 feet 28 feet
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Converse of Pythagorean Theorem

Converse of the Pythagorean Theorem as name suggests, is converse statement of Pythagorean Theorem. Pythagorean Theorem itself states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The converse of this theorem flips the logic. if a triangle has sides such that the square of the longest side is equal to the sum of the squares of the other two sides, then the triangle must be a right triangle.

In this article, we are going to discuss the converse of the Pythagorean theorem, its proof, and some solved examples based on them.

What is the Pythagorean Theorem?

The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite to the right angle) is equal to the sum of the squares of the other two sides.

Mathematically, Pythagorean theorem can be stated as. a 2 + b 2 = c 2 Where. a and b are other two sides of triangle and, c is hypotenuse of right triangle.

What is the Converse of the Pythagorean Theorem?

The converse of the Pythagorean 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 lengths of the other two sides, then the triangle is a right-angled triangle. It is simply the reverse of the original Pythagorean theorem. In simpler terms, if we are given a triangle and we know the length of all its three sides, we can check if it is a right-angled triangle or not by using the converse rule.

For Example. Let’s suppose we are given an triangle with sides 5 cm, 12 cm, and 13 cm. We have to check if this is a right-angled triangle. We will use the converse of the Pythagorean Theorem to check.

First, we square the lengths of the sides.

Now we check if sum of square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides or not.

169 = 25 + 144

Since, the squares add up correctly, so given triangle is a right-angled triangle.

Proof of Converse of Pythagorean Theorem

We are going to see the proof of converse of Pythagorean theorem.

Statement. If in a triangle, the square of one side is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

Given Triangles in above Image.

Triangle ABC with ∠ACB = 90°

Triangle PQR with ∠PQR = 90°

Now, we identify the sides.

In triangle ABC. Hypotenuse is AB and the other two sides is AC and BC

In triangle PQR. Hypotenuse is PR and other two sides PQ and QR

Now, we know the Pythagorean theorem for each triangle.

For triangle ABC. AB 2 =AC 2 + BC 2

For triangle PQR. PR 2 = PQ 2 + QR 2

now, To prove the converse, let’s assume that in triangle ABC.

AB 2 = AC 2 + BC 2

And in triangle PQR.

PR 2 = PQ 2 + QR 2

Since the square of the hypotenuse is equal to the sum of the squares of the other two sides in both triangles, the converse of the Pythagorean theorem states that these triangles must be right triangles.

Since triangles ABC and PQR satisfy the condition of the converse of the Pythagorean theorem, it is proven that they are right triangles with ∠ACB = 90° and ∠PQR = 90°, respectively.

Thus, the converse of the Pythagorean theorem is proven using the given triangles.

Solved Examples on Converse of the Pythagorean Theorem

Examples 1. Given a triangle ABC which has sides of lengths 7 cm, 24 cm, and 25 cm. Determine it if is a right-angled triangle or not.

Given sides of length. 7 cm, 24 cm, and 25 cm Square the sides. 7 2 = 49 24 2 = 576 25 2 = 625 Now, we check if the square of longest side is equal to sum of other two sides. 49 + 576 = 625 Since 625 = 625, triangle ABC is a right-angled triangle.

Examples 2. A triangle has sides 5 cm, 12 cm, and 14 cm. Use the converse of the Pythagorean Theorem to check if it’s right-angled or not.

Given sides of length. 5 cm, 12 cm, and 14 cm Square the sides. 5 2 = 25 12 2 = 144 14 2 = 196 Check. 25 + 144 = 169 169 ≠ 196 Since 169 ≠ 196, this triangle is not right-angled.

Examples 3. Given a triangle ABC which has sides of lengths 11 cm, 60 cm, and 61 cm. Determine if it is a right-angled triangle or not.

Given sides of length. 11 cm, 60 cm, and 61 cm Square the sides. 11 2 = 121 60 2 = 3600 61 2 = 3721 Now, we check if the square of the longest side is equal to the sum of the other two sides. 121 + 3600 = 3721 Since 3721 = 3721, triangle ABC is a right-angled triangle.

Examples 4. A triangle has sides 16 cm, 30 cm, and 34 cm. Use the Converse of the Pythagorean Theorem to check if it’s right-angled or not.

Given sides of length. 16 cm, 30 cm, and 34 cm Square the sides. 16 2 = 256 30 2 = 900 34 2 = 1156 Check. 256+900=1156 Since 1156=1156, this triangle is a right-angled triangle.

Practice Questions on Converse of Pythagorean Theorem

Q 1. A triangle has sides 9 cm, 12 cm, and 15 cm. Use the Converse of the Pythagorean Theorem to check if it’s right-angled or not.

Q 2. Given a triangle with sides of lengths 8 cm, 15 cm, and 17 cm. Determine if it is a right-angled triangle.

Q 3. A triangle has sides 10 cm, 24 cm, and 26 cm. Use the Converse of the Pythagorean Theorem to check if it’s right-angled or not.

Q 4. Given a triangle with sides of lengths 14 cm, 48 cm, and 50 cm. Determine if it is a right-angled triangle.

Q 5. A triangle has sides 12 cm, 16 cm, and 20 cm. Use the Converse of the Pythagorean Theorem to check if it’s right-angled or not.

Q 6. A triangle has sides 5 cm, 8 cm, and 10 cm. Determine if it is a right-angled triangle or not.

Q 7. Given a triangle with sides of lengths 21 cm, 28 cm, and 35 cm. Determine if it is a right-angled triangle.

Q 8. A triangle has sides 15 cm, 20 cm, and 25 cm. Use the Converse of the Pythagorean Theorem to check if it’s right-angled or not.

Q 9. Given a triangle with sides of lengths 18 cm, 24 cm, and 30 cm. Determine if it is a right-angled triangle.

Q 10. A triangle has sides 10 cm, 20 cm, and 22 cm. Use the Converse of the Pythagorean Theorem to check if it’s right-angled or not.

1. Right-angled

2. Right-angled

3. Right-angled

4. Right-angled

5. Right-angled

6. Not right-angled

7. Right-angled

8. Right-angled

9. Right-angled

10. Not right-angled

The converse of the Pythagorean theorem is reverse theorem of Pythagorean theorem. It is an important concept of geometry mathematics. It is important to understand how we can apply this concept. We can use this concept to determine whether a triangle is right-angled if all its three sides are given. We should practice different questions based on this concept to understand this concept much better.

• Pythagoras Theorem
• Pythagorean Theorem formula
• Practice Questions on Pythagoras Theorem

FAQs on Converse of Pythagorean Theorem

The Pythagorean theorem states that in a given right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides of triangle.

The converse of the Pythagorean 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 lengths of the other two sides, then the triangle is a right-angled triangle.

How is the Converse of the Pythagorean Theorem used in geometry?

It is used to identify whether a triangle is right-angled when we are given the length all three sides of triangle.

What is the formula of converse of the Pythagorean theorem?

The formula for converse of the Pythagorean theorem is a 2 = b 2 + c 2 where a is the side in front of right triangle (known as hypotenuse), b and c are other two sides of triangle.

Why is the converse of the Pythagorean theorem useful?

The converse of the Pythagorean theorem helps us to determine whether a triangle is right-angled or not if all its three sides are given. It has various real -life applications in various fields.

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