Deck 20: Similarity

Question 1

Are the two shaded triangles below similar? If so, then what is the scale factor? If not, why are they not similar?

Answer 1

They are similar, with scale factor 1.6 (or 0.625).

Question 2

Frederick drew a quadrilateral on the chalkboard. He then wanted to draw a new quadrilateral similar to the original one, with a scale factor of 3. This is what he did:When he was done, Frederick was confused because the new quadrilateral did not appear to be "three times as large" as the original.
When he was done, Frederick was confused because the new quadrilateral did not appear to be "three times as large" as the original.

A) What did Frederick do incorrectly?
B) What is the scale factor for the two figures he did draw?

Answer 2

A) Fredrick forgot to measure the 3 feet from the center.
B) The scale factor in his drawing is 4, not 3.

Question 3

Equilateral triangle X has sides 8 centimeters long, and equilateral triangle Y has sides 12 centimeters long. Are X and Y similar? Explain.

Answer 3

Yes. Since the triangles are equilateral, all three pairs of corresponding angles will have the same size (60°). This fact assures that the 8:12 ratios are all equal, but that fact can be ascertained directly, knowing that the lengths of the sides of an equilateral triangle are all the same.

Question 4

Complete the following sentences.

A) Eight feet is _____ times as long as six feet.
B) Eight feet is _____ times longer than six feet.

Question 5

Complete the following sentences.

A) A length of 3.8 centimeters is _____ times longer than 1.1 centimeters.
B) A length of 24 inches is _____% as long as 36 inches.
C) A change in the number of graduates from Smalltown High from 20 to 27 represents a _____% increase.

Question 6

Every two right triangles are similar.

Question 7

A) Explain why triangles ACD and AEB in the figure must be similar.
B) Knowing that triangles ACD and AEB are similar, find x and y in the figure.

Question 9

Triangle ABC is similar to triangle DEC, with E corresponding to B. If you know that AC = 6 cm in length, AB = 4 cm, CD = 10 cm, and CE = 13 cm, find the lengths of the other segments and the measures of the angles not given. Be sure to give units.

A) Length of BC = _____
B) Measure of angle BCA = _____
C) Measure of angle D = _____
D) Length of DE = _____
E) Measure of angle ECD = _____
F) Measure of angle E = _____

Question 10

A rectangle 10 centimeters by 12 centimeters (width by length) is the image of a shape having length 4 centimeters, for some size change.

A) What is the scale factor of the size transformation?
B) What are the dimensions of the original shape?
C) Are the two quadrilaterals similar? Explain.

Question 11

Suppose a segment is 8.25 cm long.

A) How long is a segment 2times as long as this segment?
B) How long is a segment 2 times longer than this segment? .
C) How long is a segment 300% longer than this segment?

Question 12

Complete each statement.

A) _____ is 25% of 200.
B) _____ is 200% as much as 20.
C) Thirty is 1 $\frac { 1 } { 2 }$ times more than _____.

Question 13

The ruler method was used to enlarge the right triangle in the figure. What is the scale factor? What is x? (Note: This figure is not drawn to scale.)

Question 14

In map A, the distance from San Diego to Las Vegas is represented as 1.5 inches. The actual distance is 350 miles. In map B, the distance from San Diego to Las Vegas is represented as 2 inches, and the distance between Las Vegas and Ely, Nevada, is represented as 3 inches. What is the actual distance between Las Vegas and Ely? Write enough down to explain your thinking. (Hint: A diagram may help. Don't forget the units.)

Question 15

Suppose triangle ABC is similar to triangle DEF with AB = 6, BC = 5, AC = 7, and DE = 10, all in centimeters. Assume $\angle A = \angle D , \angle B = \angle E , \angle C = \angle F$

A) Find the lengths of DF and EF.
B) If the area of triangle ABC = about 15 cm², what is the area of triangle DEF? Justify your answer.

Question 16

Complete each statement.

A) Kristin charges 25% less than what Beth charges for tutoring. Beth charges $30.00, so Kristin charges $_____.
B) Twelve seconds is as much as _____ seconds.
C) A 48° angle is 50% larger than a _____ degree angle.

Question 17

Suppose a segment is 5 inches long.

A) How long is a segment that is 3 times as long as this segment?
B) How long is a segment that is 2 times longer than this segment?
C) How long is a segment that is125% longer than the segment originally given in part A

Question 18

Explain why the following shapes are similar or are not similar.

A) Every two triangles.
B) Every two squares.

Question 19

Triangle A has an area of 50 cm². Triangle B is similar to Triangle A but has sides twice as long. What is the area of Triangle B?

Question 20

The ruler method was used to shrink the right triangle in the figure. What is the scale factor for the length of the two triangles? What is x? (Note: The figures are not drawn to scale.)

Question 21

Given the following picture, expand it using a scale factor of 2 and the given center.

Question 22

The two shapes below are similar. Find the missing angle size, y, and the missing length, x. Justify your answers.

Question 23

The triangles are deliberately not drawn to correct scale.

A) How do you know that the two triangles are similar?
B) Find x and y. (Show your work for credit.)

Question 24

In a photograph of two buildings, building P is 5 centimeters tall and building Q is 7 centimeters tall. In an enlargement of the photograph, building P is 8 centimeters tall. How tall is building Q in the enlargement?

A) 10 centimeters
B) 11 $\frac { 1 } { 5 }$ Centimeters
C) 4 $\frac { 3 } { 8 }$ centimeters
D) There is not enough information to answer the question.

Question 25

What is the length of x, to the nearest tenth centimeter?

A) 1.9 centimeters
B) 5.1 centimeters
C) 0.3 centimeters
D) 1.3 centimeters
E) There is not enough information to answer the question.

Question 26

What is the length of y, to the nearest tenth centimeter?

A) 6.5 centimeters
B) 4.2 centimeters
C) 2.2 centimeters
D) 5.8 centimeters
E) There is not enough information to answer the question.

Question 27

Refer to the similar figures below. What is the length of x, to the nearest tenth centimeter?

A) 3.9 centimeters
B) 12.6 centimeters
C) 11 centimeters
D) 12 centimeters
E) 14.4 centimeters

Question 28

Two pentagons have the following lengths of sides.
Pentagon A: 7 centimeters, 7 centimeters, 7 centimeters, 7 centimeters, 10 centimeters
Pentagon B: 4 centimeters, 4 centimeters, 4 centimeters, 4 centimeters, 7 centimeters
Are Pentagons A and B similar? Explain your decision.

Question 29

Triangle A and triangle B are similar. The area of triangle A is 560 square centimeters. The area of triangle B is 35 square centimeters. How long would a segment in triangle A be if the corresponding segment in triangle B is 4 centimeters?

A) 64 centimeters
B) 16 centimeters
C) 4 centimeters
D) 1 centimeters
E) There is not enough information to answer the question.

Question 30

Susan is 166 centimeters tall, and in the afternoon she casts a shadow about 55 centimeters long. Her sister, who is standing next to her, casts a shadow about 35 centimeters long. About how tall is Susan's sister (to the nearest centimeter)?

A) 146 centimeters
B) 106 centimeters
C) 95 centimeters
D) 90 centimeters
E) 12 centimeters

Question 31

A triangle is similar to a larger triangle that has lengths three times as long as the corresponding lengths in the smaller triangle. How large is the angle in the smaller triangle that corresponds to a 30° angle in the larger triangle?

A) 10°
B) 30°
C) 90°
D) 270°
E) None of the answers is correct.

Question 32

What would you have to do to determine whether another quadrilateral EFGH (not shown) is similar to ABCD in the figure?

Question 33

Treat the measurements in the drawing below as accurate, even though the drawings are intentionally "off." Are the two quadrilaterals similar? Explain.

Question 34

You want to make a larger, similar version of the trapezoid below so that the 4-centimeter side will be 10 centimeters in the larger version.

A) How long will the other three sides of the larger trapezoid be?
B) What are the sizes of the angles in the larger version?

Question 35

If the two polyhedra below are similar and oriented alike, then what are the values of x and y rounded to the nearest tenth?

Question 36

Two regular pentagonal pyramids have the following dimensions.

$\begin{array} { l l l } & \text { Length of base edge } & \text { Length of lateral edge } \\\text { Pyramid A: } & 7 \text { centimeters } & 20 \text { centimeters } \\\text { Pyramid B: } & 4 \text { centimeters } & 7 \text { centimeters }\end{array}$
Are the two pyramids (mathematically) similar? Explain your reasoning.

Question 37

A right rectangular prism has a volume of 2 cubic inches. A second right rectangular prism is similar to the first one and has a volume of 128 cubic inches.

A) What is the scale factor to go from the first prism to the second?
B) What is the scale factor to go from the second prism to the first?

Question 38

If a cube has edges that are 2 centimeters long, what is the total surface area of a larger similar cube with a scale factor of 2.3?

Question 39

Suppose you have an L-shaped figure such as the one below. What is the volume of a figure similar to this one rounded to the nearest tenth, with a scale factor of 2.8? Each cube has a volume of 1 cubic unit.

Question 40

Each face of the small cubes making the shape below has an area of 1 square unit.

A)
Surface area: _____
Volume: _____

B) Suppose another prism is similar to the one above but two times larger in all dimensions. What is the surface area and volume of the new prism? Write enough to make your thinking clear

Question 41

A right rectangular prism with dimensions $\frac { 2 } { 3 }$ by $\frac { 4 } { 3 }$ by $\frac { 8 } { 3 }$ (each in inches) is enlarged by a scale factor of $1 \frac { 1 } { 2 }$

A) The volume of the prism is _____.
B) The surface area of the original prism is _____ times as much as the surface area of the larger right rectangular prism.

Question 42

Jeff's hot tub is 3 feet deep and holds 1500 gallons of water. He's building a pool that is similar in shape but 9 feet deep. How much water will his pool hold?

Question 43

Explain why or why not the following shapes are similar (or not similar).

A) every two rectangular prisms
B) every two cubes
C) every two pyramids so long as the base is a square region

Question 44

Pinocchio's nose grew to four times its normal length. If the rest of Pinocchio's body grew in the same way, what would be his new surface area?

A) 4 times his original surface area
B) 8 times his original surface area
C) 16 times his original surface area
D) 64 times his original surface area
E) None of the answers is correct.

Question 45

A teacher has a model of an L-shaped building made of four cubes that snap together. She wants to make an enlargement, similar in shape to the original. She has only 600 cubes. What scale factor should she use to make the largest building possible? Explain your reasoning.

Question 46

Consider the shape in the figure, made of centimeter cubes glued together face to face. What is the volume of the shape, in cubic centimeters?

A) 11
B) 10
C) 8
D) 6
E) 5

Question 47

Consider the shape in the figure, made of centimeter cubes glued together face to face. A larger shape, similar to the one below, is also made of cubes. The larger shape is twice as tall, twice as wide, twice as deep, and in general twice as long in every corresponding length dimension. What is the volume of the larger shape, in cubic centimeters?

A) 40
B) 20
C) 16
D) 10
E) None of the answers is correct.