Geometry Review: Special Right Triangles

For Use With Lesson 13-2

In Geometry, you learned about two special right triangles, the 45° − 45 ° − 90 ° negative 45 degrees negative 90 degrees  triangle and the 30° − 60 ° − 90 ° negative 60 degrees negative 90 degrees  triangle. The figures below summarize the relationships among the lengths of the sides of each triangle.

Image Long Description

Example 1

Find the missing side lengths in each 45° − 45 ° − 90 ° negative 45 degrees negative 90 degrees  triangle.

A		B
y = 2 ⋅ 5 y = 5 2 table with 2 rows and 1 column , row1 column 1 , y equals square root of 2 dot 5 , row2 column 1 , y equals 5 square root of 2 , end table	hypotenuse = 2 · leg Simplify . table with 2 rows and 1 column , row1 column 1 , hypotenuse . equals square root of 2 middle dot leg , row2 column 1 , cap simplify , . , end table	5 = 2 ⋅ x x = 5 2 = 5 2 2 table with 2 rows and 1 column , row1 column 1 , 5 equals square root of 2 dot x , row2 column 1 , x equals , fraction 5 , over square root of 2 end fraction , equals , fraction 5 square root of 2 , over 2 end fraction , end table

Example 2

Find the missing side lengths in the 30 ° − 60 ° − 90 ° 30 degrees negative 60 degrees negative 90 degrees  triangle below.

4 = 3 ⋅ x longer leg = 3 ⋅ shorter leg x = 4 3 = 4 3 3 Divide and simplify . y = 2 x hypotenuse = 2 ⋅ shorter leg y = 2 ⋅ 4 3 3 = 8 3 3 Substitute 4 3 3 for  x  and simplify . table with 4 rows and 2 columns , row1 column 1 , 4 equals square root of 3 dot x , column 2 longerleg . equals square root of 3 dot . shorterleg , row2 column 1 , x equals , fraction 4 , over square root of 3 end fraction , equals , fraction 4 square root of 3 , over 3 end fraction , column 2 cap divideandsimplify . . , row3 column 1 , y equals 2 x , column 2 hypotenuse . equals 2 dot . shorterleg , row4 column 1 , y equals 2 dot , fraction 4 square root of 3 , over 3 end fraction , equals , fraction 8 square root of 3 , over 3 end fraction , column 2 cap substitute . fraction 4 square root of 3 , over 3 end fraction . for , x . andsimplify . . , end table

Exercises

Use the given information to find the missing side length(s) in each 45° − 45 ° − 90 ° negative 45 degrees negative 90 degrees  triangle. Rationalize any denominators.

1. hypotenuse 1 in.
2. leg 2 cm
3. hypotenuse 3 square root of 3  ft
4. leg 2 5 2 square root of 5  m

Use the given information to find the missing side lengths in each 30 ° − 60 ° − 90 ° 30 degrees negative 60 degrees negative 90 degrees  triangle. Rationalize any denominators.

5. shorter leg 3 in.
6. longer leg 1 cm
7. hypotenuse 1 ft
8. shorter leg 3 square root of 3  cm

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