7-4 Similarity in Right Triangles

Quick Review

C D ¯ c d bar  is the altitude to the hypotenuse of right △ A B C . white up pointing triangle eh b c .

• △ A B C ∼ △ A C D , white up pointing triangle eh b c , tilde operator white up pointing triangle eh c d comma   △ A B C ∼ △ C B D , white up pointing triangle eh b c , tilde operator white up pointing triangle c b d comma  and △ A C D ∼ △ C B D white up pointing triangle eh c d , tilde operator white up pointing triangle c b d
• A D C D = C D D B ′ , A B A C = A C A D ′ ,  and  A B C B = C B D B fraction eh d , over c d end fraction . equals . fraction c d , over d , b prime end fraction . comma . fraction eh b , over eh c end fraction . equals . fraction eh c , over eh , d prime end fraction . comma , and . fraction eh b , over c b end fraction . equals . fraction c b , over d b end fraction

Example

What is the value of x?

5 + x 10 = 10 5 Write a proportion . 5 ( 5 + x ) = 100 Cross Products Property 25 + 5 x = 100 Distributive Property 5 x = 75 Subtract  25  from each side . x = 15 Divide each side by  5. table with 5 rows and 3 columns , row1 column 1 , fraction 5 plus x , over 10 end fraction , column 2 equals , 10 over 5 , column 3 cap writeaproportion . . , row2 column 1 , 5 . open , 5 plus x , close , column 2 equals 100 , column 3 cap crosscap productscap property , row3 column 1 , 25 plus 5 x , column 2 equals 100 , column 3 cap distributivecap property , row4 column 1 , 5 x , column 2 equals 75 , column 3 cap subtract . 25 . fromeachside . . , row5 column 1 , x , column 2 equals 15 , column 3 cap divideeachsideby . 5. , end table

Exercises

Find the geometric mean of each pair of numbers.

17. 9 and 16
18. 5 and 12

Algebra Find the value of each variable. Write your answer in simplest radical form.

7-5 Proportions in Triangles

Quick Review

Side-Splitter Theorem and Corollary

If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally. If three parallel lines intersect two transversals, then the segments intercepted on the transversals are proportional.

Triangle-Angle-Bisector Theorem

If a ray bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the other two sides of the triangle.

Example

What is the value of x?

12 15 = 9 x Write a proportion . 12 x = 135 Cross Products Property x = 11.25 Divide each side by  12. table with 3 rows and 3 columns , row1 column 1 , 12 over 15 , column 2 equals , 9 over x , column 3 cap writeaproportion . . , row2 column 1 , 12 x , column 2 equals 135 , column 3 cap crosscap productscap property , row3 column 1 , x , column 2 equals , 11.25 , column 3 cap divideeachsideby . 12. , end table

Exercises

Algebra Find the value of x.

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