6.1 Ratios, Proportions, and the Geometric Mean

Obj.: Solve problems by writing and solving proportions.

Key Vocabulary

• Ratio - If a and b are two numbers or quantities and b≠ 0, then the ratio of a to b

is a/b. The ratio of a to b can also be written as a: b. • Proportion - An equation that states that two ratios are equal is called a proportion. • Means, extremes - The numbers b and c are the means of the proportion. The numbers a and d are the extremes of the proportion.

♦♦♦♦♦KEY CONCEPT♦♦♦♦♦ A Property of Proportions 1. Cross Products Property In a proportion, the product of the extremes equals the product of the means.

If b

a = d

c where b ≠ 0 and d ≠ 0, then a • d = b • c.

3

2 =

6

4, 2 • 6 = 3 • 4, 12 = 12

• Geometric mean - The geometric mean of two positive numbers a and b is the

positive number x that satisfies x

a =

b

x. So, x² = ab and x = ab .

EXAMPLE 1 Simplify ratios Simplify the ratio. (See Table of Measures, p. 921)

a. 76 cm: 8 cm b. .in

ft

24

4

Solution

EXAMPLE 2 Use a ratio to find a dimension PAINTING You are painting barn doors. You know that the perimeter of the wall is 64 feet and that the ratio of its length to its height is 3:5. Find the area of the wall. Solution

EXAMPLE 3 Use extended ratios

ALGEBRA The measures of the angles in ▲BCD are in the extended ratio of 2: 3: 4. Find the measures of the angles. Solution

EXAMPLE 4 Solve proportions ALGEBRA Solve the proportion.

a. 4

3 =

16

x b. 1

3

x = x

2

Solution

EXAMPLE 5 Solve a real-world problem Bowling You want to find the total number of rows of boards that make up 24 lanes at a bowling alley. You know that there are 117 rows in 3 lanes. Find the total number of rows of boards that make up the 24 lanes. Solution

EXAMPLE 6 Find a geometric mean

Find the geometric mean of 16 and 48. Solution

6.1 Cont.

6.2 Use Proportions to Solve Geometry Problems

Obj.: Use proportions to solve geometry problems.

Key Vocabulary • Scale drawing - A scale drawing is a drawing that is the same shape as the object it represents. • Scale - The scale is a ratio that describes how the dimensions in the drawing are related to the actual dimensions of the object.

♦♦♦♦♦KEY CONCEPT♦♦♦♦♦ Additional Properties of Proportions 2. Reciprocal Property If two ratios are equal, then their reciprocals are also

equal. If b

a =

d

c, then

a

b =

c

d

3. If you interchange the means of a proportion, then you form another

true proportion. If b

a =

d

c, then

c

a =

d

b

4. In a proportion, if you add the value of each ratio’s denominator to its numerator, then you form another

true proportion. If b

a =

d

c, then

b

ba =

d

dc

EXAMPLE 1 Use properties of proportions

In the diagram, DF

AC = EF

BC . Write four true proportions.

Solution

EXAMPLE 2 Use proportions with geometric figures

ALGEBRA In the diagram, LH

JL =

KG

JK. Find JH and JL.

Solution

EXAMPLE 3 Find the scale of a drawing Keys The length of the scale drawing is 7 centimeters. The length of the actual key is 4 centimeters. What is the scale of the drawing? Solution

EXAMPLE 4 Use a scale drawing MAPS The scale of the map at the right is 1 inch: 8 miles. Find the actual distance from Westbrook to Cooley. Solution

EXAMPLE 5 TAKS Reasoning: Multi-Step Problem SCALE MODEL You buy a 3-D scale model of the Sunsphere in Knoxville, TN. The actual building is 266 feet tall. Your model is 20 inches tall, and the diameter of the dome on your scale model is about 5.6 inches. a. What is the diameter of the actual dome? b. About how many times as tall as your model is the actual building? Solution

6.2 Cont.

6.3 Use Similar Polygons

Obj.: Use proportions to identify similar polygons.

Key Vocabulary • Similar polygons - Two polygons are similar polygons if corresponding angles

are congruent and corresponding side lengths are proportional.

• Scale factor - If two polygons are similar, then the ratio of the lengths of

two corresponding sides is called the scale factor.

Corresponding angles

∠A ≅ ∠E, ∠B ≅ ∠F, ∠C ≅ ∠G,

and ∠D ≅ ∠H

ABCD ∼ EFGH Ratios of corresponding sides

Order Matters!! EF

AB = FG

BC = GH

CD = HE

DA

Perimeters of Similar Polygons Theorem If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths.

If KLMN ∼ PQRS, then SPRSQRPQ

NKMNLMKL

=

PQ

KL = QR

LM = RS

MN = SP

NK .

♦♦♦♦♦KEY CONCEPT♦♦♦♦♦

Corresponding Lengths in Similar Polygons If two polygons are similar, then the ratio of any two corresponding lengths in the polygons is equal to the scale factor of the similar polygons.

EXAMPLE 1 Use similarity statements

In the diagram, ▲ABC ∼ ▲DEF.

a. List all pairs of congruent angles. b. Check that the ratios of corresponding side lengths are equal. c. Write the ratios of the corresponding side lengths in a statement of proportionality. Solution

EXAMPLE 2 Find the scale factor (6.3 cont.) Determine whether the polygons are similar. If they are, write a similarity statement and find the scale factor of ABCD to JKLM. Solution

EXAMPLE 3 Use similar polygons

ALGEBRA In the diagram, ▲BCD ∼ ▲RST. Find the value of x.

Solution

EXAMPLE 4 Find perimeters of similar figures Basketball A large cement court is being poured for a basketball hoop in place of a smaller one. The court will be 20 feet wide and 25 feet long. The old court was similar in shape, but only 16 feet wide. a. Find the scale factor of the new court to the old court. b. Find the perimeter of the new court and the old court. Solution

EXAMPLE 5 Use a scale factor

In the diagram, ▲TPR ∼ ▲XPZ. Find the length of the altitude GL.

Solution

6.3 Cont.

6.4 Prove Triangles Similar by AA

Obj.: Use the AA Similarity Postulate.

Key Vocabulary • Similar polygons - Two polygons are similar polygons if corresponding angles are congruent and corresponding side lengths are proportional. Angle-Angle (AA) Similarity Postulate If two angles of one triangle are congruent to two angles of another triangle, then the

two triangles are similar. ∆JKL ~ ∆XYZ EXAMPLE 1 Use the AA Similarity Postulate Determine whether the triangles are similar. If they are, write a similarity statement. Explain your reasoning. Solution

EXAMPLE 2 Show that triangles are similar Show that the two triangles are similar. a. ∆RTV and ∆RQS b. ∆LMN and ∆NOP Solution

EXAMPLE 3 Using similar triangles

Height A lifeguard is standing beside the lifeguard chair on a beach. The lifeguard is 6 feet 4 inches tall and casts a shadow that is 48 inches long. The chair casts a shadow that is 6 feet long. How tall is the chair? Solution

6.5 Prove Triangles Similar by SSS and SAS

Obj.: Use the SSS and SAS Similarity Theorems.

Key Vocabulary • ratio, p. 356 • proportion, p. 358 • similar polygons, p. 372

Side-Side-Side (SSS) Similarity Theorem If the corresponding side lengths of two triangles are proportional, then the triangles are similar.

If

=

=

, then ∆ABC ~ ∆RST.

Side-Angle-Side (SAS) Similarity Theorem If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar.

If ∠X ≅ ∠M and

=

, then ∆XYZ ~ ∆MNP

EXAMPLE 1 Use the SSS Similarity Theorem Is either ∆DEF or ∆GHJ similar to ∆ABC? Solution

EXAMPLE 2 Use the SSS Similarity Theorem ALGEBRA Find the value of x that makes ∆ABC ~∆DEF. Solution

6.6 Use Proportionality Theorems

Obj.: Use proportions with a triangle or parallel lines.

Key Vocabulary • Corresponding angles - Two angles are corresponding angles if they have corresponding positions. For example,

∠2 and∠6 are above the lines and to the right of the transversal t. • Ratio - If a and b are two numbers or quantities and b ≠ 0, then the ratio of a to b is a/b. The ratio of a to b can also be written as a : b. • Proportion - An equation that states that two ratios are equal is called a proportion. Triangle Proportionality Theorem ∆ prop. Th If a line parallel to one side of a triangle intersects the other two sides, then it

divides the two sides proportionally. If ̅̅ ̅̅ ⫽ ̅̅̅̅ , then

=

Converse of the Triangle Proportionality Theorem If a line divides two sides of a triangle proportionally, then it is parallel to the

third side. If

=

, then ̅̅ ̅̅ ⫽ ̅̅̅̅

3 ⫽ lines inters. 2 trans. → prop. If three parallel lines intersect two transversals, then they divide the transversals proportionally.

=

ray bis. ∠ of ∆→opp. side prop. to other 2 sides If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the

lengths of the other two sides.

=

EXAMPLE 1 Find the length of a segment In the diagram, ̅̅̅̅ ⫽ ̅̅ ̅̅ , RQ = 10, RS = 12, and ST = 6,

What is the length of ̅̅ ̅̅ ? Solution

EXAMPLE 2 Solve a real-world problem

Aerodynamics A spoiler for a remote controlled car is shown where AB = 31 mm, BC = 19,

CD = 27 mm, and DE = 23 mm. Explain why ̅̅ ̅̅ is not parallel to ̅̅̅̅ . Solution EXAMPLE 3 Use Theorem 6.6 Farming A farmer’s land is divided by a newly constructed interstate. The distances shown are in meters. Find the distance CA between the north border and the south border of the farmer’s land. Solution

EXAMPLE 4 Use Theorem 6.7 In the diagram, ∠ DEG ≅ ∠ GEF. Use the given

side lengths to find the length of ̅̅ ̅̅ . Solution

6.6 Cont.