# USING THE CONVERSE OF THE PYTHAGOREAN THEOREM

## About "Using the converse of the pythagorean theorem"

Using the converse of the Pythagorean theorem :

In this section, we are going to see, how the converse of the Pythagorean theorem can be used to solve real-world problems.

## Converse of the pythagorean theorem The Pythagorean Theorem states that if a triangle is a right triangle, then, the sum of the squares of the lengths of the  legs is equal to the square of the length of the hypotenuse.

That is, if a and b are legs and c is the hypotenuse, then

a² + b²  =  c²

The converse of the Pythagorean Theorem states that if a² + b²  =  c², then the triangle is a right triangle.

## Using the converse of the pythagorean theorem - Examples

Example 1 :

Tanya is buying edging for a triangular flower garden she plans to build in her backyard. If the lengths of the three pieces of edging that she purchases are 13 feet, 10 feet, and 7 feet, will the flower garden be in the shape of a right triangle ?

Solution :

Step 1 :

Let a = 10, b = 7, and c = 13.

(Always assume the longest side as "c")

Step 2 :

Find the value of a² + b².

a² + b²  =  10² + 7²

a² + b²  =  100 + 49

a² + b²  =  149 ----- (1)

Step 3 :

Find the value of c²

c²  =  13²

=  169 ----- (2)

Step 4 :

From (1) and (2), we get

a² + b²    c²

By the converse of Pythagorean theorem, the triangle with the side lengths 13 feet, 10 feet, and 7 feet is not a right triangle.

Hence, the garden will not be in the shape of a right triangle.

Example 2 :

A blueprint for a new triangular playground shows that the sides measure 480 ft, 140 ft, and 500 ft. Is the playground in the shape of a right triangle ? Explain.

Solution :

Step 1 :

Let a = 480, b = 140, and c = 500.

(Always assume the longest side as "c")

Step 2 :

Find the value of a² + b².

a² + b²  =  480² + 140²

a² + b²  =  230,400 + 19,600

a² + b²  =  250,000 ----- (1)

Step 3 :

Find the value of c²

c²  =  500²

=  250,000 ----- (2)

Step 4 :

From (1) and (2), we get

a² + b²  =

By the converse of Pythagorean theorem, the triangle with the side lengths 480 ft, 140 ft, and 500 ft is a right triangle.

Hence, the the playground is in the shape of a right triangle.

Example 3 :

A triangular piece of glass has sides that measure 18 in., 19 in., and 25 in. Is the piece of glass in the shape of a right triangle ? Explain.

Solution :

Step 1 :

Let a = 18, b = 19, and c = 25.

(Always assume the longest side as "c")

Step 2 :

Find the value of a² + b².

a² + b²  =  18² + 19²

a² + b²  =  324 + 361

a² + b²  =  685 ----- (1)

Step 3 :

Find the value of c²

c²  =  25²

=  625 ----- (2)

Step 4 :

From (1) and (2), we get

a² + b²    c²

By the converse of Pythagorean theorem, the triangle with the side lengths 18 in., 19 in., and 25 in. is not a right triangle.

Hence, the piece of glass is not in the shape of a right triangle.

Example 4 :

A corner of a fenced yard forms a right angle. Can we place a 12 ft long board across the corner to form a right triangle for which the leg lengths are whole numbers ? Explain.

Solution :

Step 1 :

Let a and b be the legs of the triangle.

Step 2 :

Draw an appropriate diagram for the given information. Step 3 :

To form a right triangle, the legs a and b and the length of the board 12 ft must satisfy the converse of the Pythagorean theorem. That is

a² + b²  =  12²

a² + b²  =  144

But, there are no pairs of whole numbers whose squares add up to  12²  =  144.

Hence, we can not place a 12 foot long board across the corner to form a right triangle for which the leg lengths are whole numbers. After having gone through the stuff given above, we hope that the students would have understood "Using the converse of the pythagorean theorem".

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