7-1 Ratios and Proportions

Quick Review

A ratio is a comparison of two quantities by division. A proportion is a statement that two ratios are equal. The Cross Products Property states that if a b = c d , eh over b , equals , c over d , comma  where b ≠ 0 b not equal to 0  and d ≠ 0 , d not equal to 0 comma  then a d = b c . eh d equals b c .

Example

What is the solution of x x + 3 = 4 6 ? fraction x , over x plus 3 end fraction . equals , 4 sixths , question mark

6 x = 4 ( x + 3 ) Cross Products Property 6 x = 4 x + 12 Distributive Property 2 x = 12 Subtract  4 x  from each side. x = 6 Divide each side by  2 . table with 4 rows and 4 columns , row1 column 1 , 6 x , column 2 equals , column 3 4 open x plus 3 close , column 4 cap cross cap products cap property , row2 column 1 , 6 x , column 2 equals , column 3 4 x plus 12 , column 4 cap distributive cap property , row3 column 1 , 2 x , column 2 equals , column 3 12 , column 4 cap subtract . 4 x . from each side. , row4 column 1 , x , column 2 equals , column 3 6 , column 4 cap divide each side by . 2 . , end table

Exercises

5. A high school has 16 math teachers for 1856 math students. What is the ratio of math teachers to math students?
6. The measures of two complementary angles are in the ratio 2 : 3. What is the measure of the smaller angle?

Algebra Solve each proportion.

7. x 7 = 18 21 x over 7 , equals , 18 over 21
8. 6 11 = 15 2 x 6 elevenths , equals , fraction 15 , over 2 x end fraction
9. x 3 = x + 4 5 x over 3 , equals . fraction x plus 4 , over 5 end fraction
10. 8 x + 9 = 2 x − 3 fraction 8 , over x plus 9 end fraction . equals . fraction 2 , over x minus 3 end fraction

7-2 and 7-3 Similar Polygons and Proving Triangles Similar

Quick Review

Similar polygons have congruent corresponding angles and proportional corresponding sides. You can prove triangles similar with limited information about congruent corresponding angles and proportional corresponding sides.

Example

Is △ A B C white up pointing triangle eh b c  similar to △ R Q P ? white up pointing triangle r q p question mark  How do you know?

You know that ∠ A ≅ ∠ R . angle , eh approximately equal to , angle r .

A B R Q = A C R P = 2 1 , fraction eh b , over r q end fraction . equals . fraction eh c , over r p end fraction . equals , 2 over 1 , comma  so the triangles are similar by the SAS ∼ tilde operator  Theorem.

Exercises

The polygons are similar. Write a similarity statement and give the scale factor.

11. 
12. 
13. City Planning The length of a rectangular playground in a scale drawing is 12 in. If the scale is 1  in  . = 10  ft  , 1 , in , . , equals 10 , ft , comma  what is the actual length?
14. Indirect Measurement A 3-ft vertical post casts a 24-in. shadow at the same time a pine tree casts a 30-ft shadow. How tall is the pine tree?

Are the triangles similar? How do you know?

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