The Pythagorean Theorem and its Converse


Using the Pythagorean Theorem


Example

Find the value of g. Write your answer in simplest radical form.

 

Using the Pythagorean Theorem, substitute g and 9 for the legs and 13 for the hypotenuse.

 

a2 + b2 = c2

 g2 + 92 = 132           Substitute

g2 + 81 = 169           Simplify

        g2 = 88             Subtract 81 from each side

        g2 = 88           Take the square root

         g =  2(22)      Simplify

 


You can also use the Pythagorean Theorem in the other direction (that is, use the converse of the Pythagorean Theorem) to determine whether a triangle is right.

Example

Is a triangle with sides of lengths 8, 12, and 14 a right triangle?


If this is a right triangle, then the sides should follow the Pythagorean Theorem, with the longest side being the hypotenuse.
        a2 + b2 = c2

        82 + 122 ? 142

        64 + 144 ? 196

        208 > 196

The theorem doesn't hold.  So the triangle is not a right triangle.


As a bonus, however, we can figure out what kind of triangle this is.


        If a2 + b2 > c2, the triangle is acute.


        If a2 + b2 < c2, the triangle is obtuse.


Because 208 > 196, the triangle is acute.



Guided Practice

Find the missing side lengths. Leave your answers in simplest radical form.

 

1. 

 

 

 

 

 

 

2.

 

 

 

3.

 

 

 

 

 

4.

 

 

 

 

 

 

Find the value of each variable. Leave your answers in simplest radical form.

 

5.

 

 

 

 

 

6.

 

 

 

 

 

7.

 

 

 

 

8.

 

 

 

 

 

9.

 

 

 

 

 

10.

 

 

 

 

 

 

Find the area of each triangle. Leave your answers in simplest radical form.

 

11.

 

 

 

 

 

 

12.

 

 

 

 

 

 

13.

 

 

 

 

 

 

 

14.

 

 

 

 

 

 

15.

 

 

 

 

 

 

 

16.

 

 

 

 

 

 

 

17.

 

 

 

 

 

The numbers represent the lengths of the sides of a triangle. Classify each triangle as acute, obtuse, or right.

 

18.       6, 9, 10

 

 

 

 

19.       7, 24, 25

 

 

 

 

 

20.       18, 24, 30

 

 

 

 

 

21.       2, 5, 6

 

 

 

 

 

22.       20, 100, 110

 

 

 

 

 

23.       13, 21, 24








Answer Key

1.  45

2.  72

3.  8

4.  6

5.  51

6.  47

7.  265

8.  7

9.  221

10.  182

11.  420 in2

12.  2410 cm2

13.  70114 ft2

14.  468 in2

15.  1500 ft2

16.  13,000 cm2

17.  91 m2

18.  acute

19.  right

20.  right

21.  obtuse

22.  obtuse

23.  acute