Geometry: Chapter 7 Review: ANSWER KEY 1) A. …nyhsmath.weebly.com/uploads/4/2/2/9/42294853/geoch7review_answe… · Geometry: Chapter 7 Review: ANSWER KEY ... (think back to the - [PDF Document] (2024)

Geometry: Chapter 7 Review: ANSWER KEY

This answer key is incomplete as it does not show work. It is only meant to use to confirm your final results.

1) Ratios : 7.1

A. Students should know what a ratio is. (See 7.1 notes)

B. Students should know how to represent a ratio in 3 different ways. (see 7.1 notes) Ex: length of car: 14 ft 10 in. Length of model car: 8 in. Write the ratio of the length of a car to the length of the model car. 21in/1in=21/1

C. Students should understand extended ratios and know how to solve problems involving them.

Ex: A band director needs to purchase new uniforms. The ratio of small to medium to large uniforms is 3 : 4 : 6.

a. If there are 260 total uniforms to purchase, how many will be small?

3(20)=60 uniform

b. How many of these uniforms will be medium?

4(20)=80

c. How many of these uniforms will be large?

6(20)=120

2) Proportions: 7.1 A. Students should understand what a proportion is. (see 7.1 notes)

B. Students should know how to find the cross products of a proportion to solve for an unknown value.

a. 3 = 4 6

x b. 4 3 5 9

x −= c. 12 4

2 5x x=

− x=4.5 x=10.2 x=3

C. Students should be able to identify the means and extremes of a proportion and see that the proportion can represented several ways based on the properties of proportions.

Properties of Proportions

Property How to apply it

Ex. 1) a. Write a proportion that has means 4 and 15 and extremes 6 and 10.

64 =

1510

b. Write two more equivalent proportions to the one in part A.

!!"= !

!" or !

!= !"

!"

D. Students should be able to solve application problems relating to proportions. Ex. A meatloaf recipe uses 4 pounds of hamburger to feed 6 people. How many pounds of hamburger will be used to feed 15 people? 10 pounds

3) Similarity A. Students should know what similarity means (see 7.2 notes) When 2 polygons are similar: a. All of the corresponding angles are __congruent________________________

b. The ratio of the corresponding side-­‐lengths are equal

Ex 1: List all pairs of congruent angles for the figures. Then write the ratios of the corresponding sides in a statement of proportionality.

1. Δ ABC ~ Δ DEF 1. ∠𝐴 ≅ ∠𝐷,∠𝐵 ≅ ∠𝐸,∠𝐶 ≅ ∠𝐹

B. Students should be able to identify corresponding sides and angles and

evaluate whether shapes are similar. (To determine if two figures are similar, first confirm that all the angles are congruent. THEN set up the ratios of the sides of the one figure to the other and confirm all the ratios are proportional.)

EX:

Similar: Scale factor ½: Not similar.

C. Students should know how to list congruent angles, equal side ratios, and write a similarity statement.

Ex: Write a similarity statement for the similar polygons in the above example.

∆𝐴𝐵𝐶~ ∆𝐹𝐷𝐸

D. Students should know what a scale factor is and be able to find it by comparing the side of one polygon to the corresponding sides of another

𝐴𝐵𝐷𝐸 =

𝐵𝐶𝐸𝐹 =

𝐴𝐶𝐷𝐹

To find the scale factor of similar figures, find the ratio of 1 set of corresponding sides. Be sure the sides are corresponding! Ex: Find the Scale Factor: ΔLMN ~ ΔPQR Scale factor: !

!"= !

!

∆𝐴𝐵𝐶~ ∆𝐷𝐸𝐹 Scale Factor: !"!= !

!

E. Students should be able to use a scale factors to find missing side length in similar polygons. To find a missing side length, set up a proportion of corresponding sides, where one ratio of corresponding sides is the scale factor and the other is ratio is the one containing the ratio. Use cross-­‐products to find the variable. Ex: Find the value of the given variables in the similar polygons.

x = 80/3 y=6 z = 16/3 X = 4.4

F. Students should be able to solve application problems involving scale factor and similarity. (think back to the mural problem and the map problem from section 7.2) Ex. Brian bought a 3-­‐D scale model of a pool table for his desk. The length of the model is 5.6 inches long. The length of the actual pool table is 7 feet long, and the width is about 3.9 feet. a. What is the width of the model? 3.12 inches

b. About how many times as wide as the model is the actual pool table? About 15 times

4) Similarity Postulates

A. Students should know the similarity postulates (AA~, SSS~, SAS~) and understand that when comparing sides, we are not comparing one side to its corresponding side, but instead, are comparing one ratio of sides to another.

B. Students should be able to determine if triangles are similar by these postulates.

Do the triangles have to be similar? If so, write a similarity statement and tell whether you would use AA ∼ , SAS ∼ , or SSS ∼ .

a.

b.

Yes. SF=5/3. ∆𝐶𝑄𝑇~ ∆𝐶𝑀𝑃 𝑏𝑦 𝑆𝐴𝑆~ Yes. ∆𝑋𝑌𝑍~ ∆𝐸𝑁𝑊 𝑏𝑦 𝐴𝐴~

d.

e.

Not similar Yes. SF=3/2. ∆𝑄𝑅𝑃~ ∆𝑍𝑋𝑌 𝑏𝑦 𝑆𝑆𝑆~ Yes. Cross products are both 48. ∆𝐽𝐻𝐾~ ∆𝑆𝑅𝑇 𝑏𝑦 𝑆𝐴𝑆~

C. Students should be able to solve for missing sides and angles of similar triangles.

x x=6

X=4, y=3 y = 18.5

D. Students should be able to solve application problems involving similar triangles.

Ex.: 2-ft vertical post casts a 16-in. shadow at the same time a nearby cell phone tower casts a 120-ft shadow. How tall is the cell phone tower?

Ex. Explain why the triangles are similar, then find the distance across the lake. X=180ft

5) Similarity relationships within triangles. A. Students should be able to use the side splitter theorem to set up

proportions and find missing lengths.

X=10.5

B. Students should be able to use the side-splitter theorem converse to determine if lines (or planes) are parallel.

C. Students should be able to use the side-splitter theorem corollary to find missing lengths.

X= 2.4

D. Students should be able to use the triangle angle bisector theorem to find different length.

X=18 2/3

E. Students should know that the triangle angle bisector theorem is directly related to the side-splitter theorem. An angle bisector of a triangle divides the opposite side of the triangle into segments 5 cm and 3 cm long. A second side of the triangle is 7.5 cm long. Find all possible lengths for the third side of the triangle 12.5cm or 4.5 cm

F. Students should be able to solve application problems relating to the above listed theorems.

Ex. The figure below shows the locations of a high school, a computer store, a library, and a convention center. The street along which the computer store and library are located bisects the obtuse angle formed by two of the other streets. Use the information in the figure to find the distance from the library to the convention center.

4.5 miles from the library to convention center Ex.

x= 79.83