write and simplify ratios. use proportions to solve problems. objectives

Write and simplify ratios.

Use proportions to solve problems.

Objectives

ratioproportionextremesmeanscross products

Vocabulary

A ratio compares two numbers by division. The ratio

of two numbers a and b can be written as a to b, a:b,

or , where b ≠ 0. For example, the ratios 1 to 2,

1:2, and all represent the same comparison.

In a ratio, the denominator of the fraction cannot be zero because division by zero is undefined.

Remember!

Example 2: Using Ratios

The ratio of the side lengths of a triangle is 4:7:5, and its perimeter is 96 cm. What is the length of the shortest side?

Let the side lengths be 4x, 7x, and 5x. Then 4x + 7x + 5x = 96 . After like terms are combined, 16x = 96. So x = 6. The length of the shortest side is 4x = 4(6) = 24 cm.

Check It Out! Example 2

The ratio of the angle measures in a triangle is 1:6:13. What is the measure of each angle?

x + y + z = 180°

x + 6x + 13x = 180°

20x = 180°

x = 9°

y = 6x

y = 6(9°)

y = 54°

z = 13x

z = 13(9°)

z = 117°

A proportion is an equation stating that two ratios

are equal. In the proportion , the values

a and d are the extremes. The values b and c

are the means. When the proportion is written as

a:b = c:d, the extremes are in the first and last

positions. The means are in the two middle positions.

In Algebra 1 you learned the Cross Products Property. The product of the extremes ad and the product of the means bc are called the cross products.

The following table shows equivalent forms of the Cross Products Property.

Warm UpSolve each proportion.

1. 2. 3.

z = ±10 x = 8

Figures that are similar (~) have the same shape but not necessarily the same size.

Two polygons are similar polygons if and only if their corresponding angles are congruent and their corresponding side lengths are proportional.

Check It Out! Example 1

Identify the pairs of congruent angles and corresponding sides.

B G and C H. By the Third Angles Theorem, A J.

A similarity ratio is the ratio of the lengths of

the corresponding sides of two similar polygons.

The similarity ratio of ∆ABC to ∆DEF is , or .

The similarity ratio of ∆DEF to ∆ABC is , or 2.

Writing a similarity statement is like writing a congruence statement—be sure to list corresponding vertices in the same order.

Writing Math

Example 2B: Identifying Similar Polygons

Determine whether the polygons are similar. If so, write the similarity ratio and a similarity statement.

∆ABCD and ∆EFGH

Example 2B Continued

Step 1 Identify pairs of congruent angles.

P R and S W isos. ∆

Step 2 Compare corresponding angles.

Since no pairs of angles are congruent, the triangles are not similar.

mW = mS = 62°

mT = 180° – 2(62°) = 56°

Check It Out! Example 2

Step 1 Identify pairs of congruent angles.

Determine if ∆JLM ~ ∆NPS. If so, write the similarity ratio and a similarity statement.

N M, L P, S J

Check It Out! Example 2 Continued

Step 2 Compare corresponding sides.

Thus the similarity ratio is , and ∆LMJ ~ ∆PNS.

When you work with proportions, be sure the ratios compare corresponding measures.

Helpful Hint

There are several ways to prove certain triangles are similar. The following postulate, as well as the SSS and SAS Similarity Theorems, will be used in proofs just as SSS, SAS, ASA, HL, and AAS were used to prove triangles congruent.

Example 2A: Verifying Triangle Similarity

Verify that the triangles are similar.

∆PQR and ∆STU

Therefore ∆PQR ~ ∆STU by SSS ~.

A A by Reflexive Property of , and B C since they are both right angles.

Example 3: Finding Lengths in Similar Triangles

Explain why ∆ABE ~ ∆ACD, and then find CD.

Step 1 Prove triangles are similar.

Therefore ∆ABE ~ ∆ACD by AA ~.

Check It Out! Example 3

Explain why ∆RSV ~ ∆RTU and then find RT.

Step 1 Prove triangles are similar.

It is given that S T. R R by Reflexive Property of .

Therefore ∆RSV ~ ∆RTU by AA ~.