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 = Take the square root
g = 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.
2.
3. 8
4. 6
5.
6.
7.
8.
9.
10.
11. 420 in2
12. cm2
13. 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