8-1 The Pythagorean Theorem and Its Converse

Quick Review

The Pythagorean Theorem holds true for any right triangle.

( leg 1 ) 2 + ( leg 2 ) 2 = ( hypotenuse ) 2 a 2 + b 2 = c 2 table with 2 rows and 2 columns , row1 column 1 , open , leg sub 1 , close squared . plus . open , leg sub 2 , close squared , column 2 equals . open . hypotenuse . close squared , row2 column 1 , eh squared , plus , b squared , column 2 equals , c squared , end table

The Converse of the Pythagorean Theorem states that if a 2 + b 2 = c 2 , eh squared , plus , b squared , equals , c squared , comma where c is the greatest side length of a triangle, then the triangle is a right triangle.

Example

What is the value of x?

a 2 + b 2 = c 2 Pythagorean Theorem x 2 + 12 2 = 20 2 Substitute . x 2 = 256 Simplify . x = 16 Take the square root . table with 4 rows and 3 columns , row1 column 1 , eh squared , plus , b squared , column 2 equals , c squared , column 3 cap pythagoreancap theorem , row2 column 1 , x squared , plus , 12 squared , column 2 equals , 20 squared , column 3 cap substitute . . , row3 column 1 , x squared , column 2 equals 256 , column 3 cap simplify , . , row4 column 1 , x , column 2 equals 16 , column 3 cap takethesquareroot . . , end table

Exercises

Find the value of x. If your answer is not an integer, express it in simplest radical form.

8-2 Special Right Triangles

Quick Review

45 ° - 45 ° - 90 °  Triangle 45 degrees minus 45 degrees minus 90 degrees . cap triangle

hypotenuse  = 2 ·  leg hypotenuse . equals square root of 2 middle dot , leg

30 ° - 60 ° - 90 °  Triangle 30 degrees minus 60 degrees minus 90 degrees . cap triangle

hypotenuse  = 2 ·  shorter leg hypotenuse . equals 2 . middle dot . shorterleg

longer leg  = 3 ·  shorter leg longerleg . equals square root of 3 . middle dot . shorterleg

Example

What is the value of x?

The triangle is a 30 ° - 60 ° - 90 ° 30 degrees minus 60 degrees minus 90 degrees triangle, and x represents the length of the longer leg.

longer leg = 3 · shorter leg x = 20 3 table with 2 rows and 2 columns , row1 column 1 , longerleg , column 2 equals square root of 3 middle dot . shorterleg , row2 column 1 , x , column 2 equals 20 square root of 3 , end table

Exercises

Find the value of each variable. If your answer is not an integer, express it in simplest radical form.

9. 
10. 
11. 
12. 
13. A square garden has sides 50 ft long. You stretch a hose from one corner of the garden to another corner along the garden's diagonal. To the nearest tenth, how long is the hose?

Table of Contents