USING THE CONVERSE OF THE PYTHAGOREAN THEOREM

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

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

c²  =  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. 

So, the garden is not 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  =  250,0000 -----(1)

Step 3 :

Find the value of c². 

c²  =  500²

c²  =  250,000 -----(2)

Step 4 :

From (1) and (2), we get

a² + b²  =  c²

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

So, 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

c²  =  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. 

So, 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.

So, 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.

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