Geometry A Common Core Curriculum

Ron Larson, Laurie Boswell

Chapter 8

Similarity - all with Video Answers

Educators

Section 1

Similar Polygons

Problem 1

COMPLETE THE SENTENCE For two ? gures to be similar, the corresponding angles must be ____________, and the corresponding side lengths must be _________________

Ali Soave

Ali Soave

Numerade Educator

Problem 2

DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers

Shira Strong

Numerade Educator

Problem 3

In Exercises 3 and $4,$ ind the scale factor. Then list all pairs of congruent angles and write the ratios of the corresponding side lengths in a statement of proportionality.
$$
\triangle \mathrm{ABC} \sim \triangle \mathrm{LMN}
$$

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 4

In Exercises 3 and $4,$ ind the scale factor. Then list all pairs of congruent angles and write the ratios of the corresponding side lengths in a statement of proportionality.
$$
\mathrm{DEFG} \sim \mathrm{PQRS}
$$

Shira Strong

Numerade Educator

Problem 5

In Exercises $5-8,$ the polygons are similar. Find the value of $\mathrm{x}$ .

Ali Soave

Numerade Educator

Problem 6

In Exercises $5-8,$ the polygons are similar. Find the value of $\mathrm{x}$ .

Shira Strong

Numerade Educator

Problem 7

In Exercises $5-8,$ the polygons are similar. Find the value of $\mathrm{x}$ .

Ali Soave

Numerade Educator

Problem 8

In Exercises $5-8,$ the polygons are similar. Find the value of $\mathrm{x}$ .

Shira Strong

Numerade Educator

Problem 9

424 Chapter 8SimilarityIn Exercises 9 and 10, the black triangles are similar. Identify the type of segment shown in blue and ?nd the value of the variable

Ali Soave

Numerade Educator

Problem 10

424 Chapter 8SimilarityIn Exercises 9 and 10, the black triangles are similar. Identify the type of segment shown in blue and ?nd the value of the variable

Shira Strong

Numerade Educator

Problem 11

In Exercises 11 and $12,$ RSTU $\sim$ ABCD. Find the ratio of their perimeters.

Ali Soave

Numerade Educator

Problem 12

In Exercises 11 and $12,$ RSTU $\sim$ ABCD. Find the ratio of their perimeters.

Shira Strong

Numerade Educator

Problem 13

In Exercises $13-16,$ two polygons are similar. The perimeter of one polygon and the ratio of the corresponding side lengths are given. Find the perimeter of the other polygon.
perimeter of smaller polygon: $48 \mathrm{cm} ;$ ratio: $\frac{2}{3}$

Ali Soave

Numerade Educator

Problem 14

In Exercises $13-16,$ two polygons are similar. The perimeter of one polygon and the ratio of the corresponding side lengths are given. Find the perimeter of the other polygon.
perimeter of smaller polygon: $66 \mathrm{ft} ;$ ratio: $\frac{3}{4}$

Shira Strong

Numerade Educator

Problem 15

In Exercises $13-16,$ two polygons are similar. The perimeter of one polygon and the ratio of the corresponding side lengths are given. Find the perimeter of the other polygon.
perimeter of larger polygon: 120 yd; ratio: $\frac{1}{6}$

Ali Soave

Numerade Educator

Problem 16

In Exercises $13-16,$ two polygons are similar. The perimeter of one polygon and the ratio of the corresponding side lengths are given. Find the perimeter of the other polygon.
perimeter of larger polygon: $85 \mathrm{m} ;$ ratio: $\frac{2}{3}$

Shira Strong

Numerade Educator

Problem 17

MODELING WITH MATHEMATICS A school gymnasium is being remodeled. The basketball court will be similar to an NCAA basketball court, which has a length of 94 feet and a width of 50 feet. The school plans to make the width of the new court 45?feet. Find the perimeters of an NCAA court and of the new court in the school.

Ali Soave

Numerade Educator

Problem 18

MODELING WITH MATHEMATICS Your family has decided to put a rectangular patio in your backyard, similar to the shape of your backyard. Your backyard has a length of 45 feet and a width of 20 feet. The length of your new patio is 18 feet. Find the perimeters of your backyard and of the patio.

Shira Strong

Numerade Educator

Problem 19

In Exercises $19-22,$ the polygons are similar. The area of one polygon is given. Find the area of the other polygon.

Jerelyn Nevil

Numerade Educator

Problem 20

In Exercises $19-22,$ the polygons are similar. The area of one polygon is given. Find the area of the other polygon.

Shira Strong

Numerade Educator

Problem 21

In Exercises $19-22,$ the polygons are similar. The area of one polygon is given. Find the area of the other polygon.

Ali Soave

Numerade Educator

Problem 22

In Exercises $19-22,$ the polygons are similar. The area of one polygon is given. Find the area of the other polygon.

Shira Strong

Numerade Educator

Problem 23

ERROR ANALYSIS Describe and correct the error in ? nding the perimeter of triangle B. The triangles are similar.

Ali Soave

Numerade Educator

Problem 24

ERROR ANALYSIS Describe and correct the error in ? nding the area of rectangle B. The rectangles are similar.

Shira Strong

Numerade Educator

Problem 25

In Exercises 25 and $26,$ decide whether the red and blue polygons are similar.

Ali Soave

Numerade Educator

Problem 26

In Exercises 25 and $26,$ decide whether the red and blue polygons are similar.

Ali Soave

Numerade Educator

Problem 27

REASONING Triangles ABC and DEF are similar. Which statement is correct? Select all that apply.

Ali Soave

Numerade Educator

Problem 28

ANALYZING RELATIONSHIPS In Exercises 28–34, JKLM ? EFGH.
Find the scale factor of JKLMto EFGH.

Shira Strong

Numerade Educator

Problem 29

ANALYZING RELATIONSHIPS In Exercises 28–34, JKLM ? EFGH.
Find the scale factor of EFGH to JKLM.

Ali Soave

Numerade Educator

Problem 30

ANALYZING RELATIONSHIPS In Exercises 28–34, JKLM ? EFGH.
Find the values of $\mathrm{x}, \mathrm{y},$ and $\mathrm{z}$

Shira Strong

Numerade Educator

Problem 31

ANALYZING RELATIONSHIPS In Exercises 28–34, JKLM ? EFGH
Find the perimeter of each polygon.

Ali Soave

Numerade Educator

Problem 32

ANALYZING RELATIONSHIPS In Exercises 28–34, JKLM ? EFGH
Find the ratio of the perimeters of IKLMto EFGH.

Shira Strong

Numerade Educator

Problem 33

ANALYZING RELATIONSHIPS In Exercises 28–34, JKLM ? EFGH
Find the area of each polygon.

Ali Soave

Numerade Educator

Problem 34

ANALYZING RELATIONSHIPS In Exercises 28–34, JKLM ? EFGH
Find the ratio of the areas of $\mathrm{fKLM}$ to EFGH.

Shira Strong

Numerade Educator

Problem 35

USING STRUCTURE Rectangle A is similar to rectangle B. Rectangle A has side lengths of 6 and 12. Rectangle B has a side length of 18. What are the possible values for the length of the other side of rectangle B? Select all that apply.

Ali Soave

Numerade Educator

Problem 36

DRAWING CONCLUSIONS In table tennis, the table is a rectangle 9 feet long and 5 feet wide. A tennis court is a rectangle 78 feet long and 36 feet wide. Are the two surfaces similar? Explain. If so, ? nd the scale factor of the tennis court to the table.

Shira Strong

Numerade Educator

Problem 37

MATHEMATICAL CONNECTIONS In Exercises 37 and 38, the two polygons are similar. Find the values of x and y

Ali Soave

Numerade Educator

Problem 38

MATHEMATICAL CONNECTIONS In Exercises 37 and 38, the two polygons are similar. Find the values of x and y

Shira Strong

Numerade Educator

Problem 39

ATTENDING TO PRECISION In Exercises 39–42, the ? gures are similar. Find the missing corresponding side?length

Ali Soave

Numerade Educator

Problem 40

ATTENDING TO PRECISION In Exercises 39–42, the ? gures are similar. Find the missing corresponding side?length

Shira Strong

Numerade Educator

Problem 41

ATTENDING TO PRECISION In Exercises 39–42, the ? gures are similar. Find the missing corresponding side?length
Figure A has an area of 48 square feet and one of the side lengths is 6 feet. Figure $B$ has an area of
75 square feet.

Ali Soave

Numerade Educator

Problem 42

ATTENDING TO PRECISION In Exercises 39–42, the ? gures are similar. Find the missing corresponding side?length
Figure A has an area of 18 square feet. Figure $\mathrm{B}$ has an area of 98 square feet and one of the side lengths is 14 l feet.

Shira Strong

Numerade Educator

Problem 43

CRITICAL THINKING In Exercises 43–48, tell whether the polygons are always, sometimes, or never similar.
two isosceles triangles

Ali Soave

Numerade Educator

Problem 44

CRITICAL THINKING In Exercises 43–48, tell whether the polygons are always, sometimes, or never similar.
two isosceles trapezoids

Shira Strong

Numerade Educator

Problem 45

CRITICAL THINKING In Exercises 43–48, tell whether the polygons are always, sometimes, or never similar.
two rhombuses

Ali Soave

Numerade Educator

Problem 46

CRITICAL THINKING In Exercises 43–48, tell whether the polygons are always, sometimes, or never similar.
two squares

Shira Strong

Numerade Educator

Problem 47

CRITICAL THINKING In Exercises 43–48, tell whether the polygons are always, sometimes, or never similar.
two regular polygons

Ali Soave

Numerade Educator

Problem 48

CRITICAL THINKING In Exercises 43–48, tell whether the polygons are always, sometimes, or never similar.
a right triangle and an equilateral triangle

Shira Strong

Numerade Educator

Problem 49

MAKING AN ARGUMENT Your sister claims that when the side lengths of two rectangles are proportional, the two rectangles must be similar. Is she correct? Explain your reasoning.

Ali Soave

Numerade Educator

Problem 50

HOW DO YOU SEE IT? You shine a ?ashlight directly on an object to project its image onto a parallel screen. Will the object and the image be similar? Explain your reasoning.

Ali Soave

Numerade Educator

Problem 51

MODELING WITH MATHEMATICS During a total eclipse of the Sun, the moon is directly in line with the Sun and blocks the Sun’s rays. The distance DA between Earth and the Sun is 93,000,000 miles, the distance DE between Earth and the moon is 240,000 miles, and the radius AB of the Sun is 432,500 miles. Use the diagram and the given measurements to estimate the radius EC of the moon

Ali Soave

Numerade Educator

Problem 52

PROVING A THEOREM Prove the Perimeters of Similar Polygons Theorem (Theorem 8.1) for similar rectangles. Include a diagram in your proof

Ali Soave

Numerade Educator

Problem 53

PROVING A THEOREM Prove the Areas of Similar Polygons Theorem (Theorem 8.2) for similar rectangles. Include a diagram in your proof.

Ali Soave

Numerade Educator

Problem 54

THOUGHT PROVOKING The postulates and theorems in this book represent Euclidean geometry. In spherical geometry, all points are points on the surface of a sphere. A line is a circle on the sphere whose diameter is equal to the diameter of the sphere. A plane is the surface of the sphere. In spherical geometry, is it possible that two triangles are similar but not congruent? Explain your reasoning.

Charles Carter

Numerade Educator

Problem 55

CRITICAL THINKING In the diagram, PQRS is a square, and PLMS ? LMRQ. Find the exact value of x. This value is called the golden ratio. Golden rectangles have their length and width in this ratio. Show that the similar rectangles in the diagram are golden rectangles.

Ali Soave

Numerade Educator

Problem 56

MATHEMATICAL CONNECTIONS The equations of?the lines shown are $y=\frac{4}{3} x \qquad 4$ and $y=\frac{4}{3} x-8 .$ Show that $\Delta A O B \sim \triangle C O D$ .

Shira Strong

Numerade Educator

Problem 57

Find the value of $\mathrm{x}$ .

Ali Soave

Numerade Educator

Problem 58

Find the value of $\mathrm{x}$ .

Shira Strong

Numerade Educator

Problem 59

Find the value of $\mathrm{x}$ .

Ali Soave

Numerade Educator

Problem 60

Find the value of $\mathrm{x}$ .

Shira Strong

Numerade Educator