Similar Figures

Show your work for each problem.

1.
Two figures are similar if they have the same _____ but not necessarily the same _____.
1. a)shape, size
2. b)area, perimeter
3. c)color, shape
4. d)size, shape
Answer: shape, size
Similar figures have the same shape (same angles, proportional sides) but can be different sizes. One figure is a scaled version of the other.
2.
Triangle ABC has sides 2, 3, and 4 cm. Triangle DEF has sides 4, 6, and 8 cm. What is the scale factor from ABC to DEF?
1. a)4
2. b)0.5
3. c)2
4. d)3
Answer: 2
Scale factor = . Each side of DEF is 2 times the corresponding side of ABC.
3.
If two similar rectangles have a scale factor of 3, and the smaller rectangle has a width of 4 cm, what is the width of the larger rectangle in cm?
Answer: 12
Width of larger rectangle = cm. The scale factor tells us how many times larger each dimension is.
4.
Which of the following pairs of shapes are ALWAYS similar?
1. a)Two rectangles
2. b)Two triangles
3. c)Two squares
4. d)Two parallelograms
Answer: Two squares
All squares have four 90-degree angles and four equal sides. This means any two squares will always have matching angles and proportional sides, making them similar.
5.
A map has a scale of 1:50000. If two cities are 3 cm apart on the map, what is the actual distance in km?
Answer: 1.5
cm = m = km. The scale factor of 50000 means 1 cm on the map represents 50000 cm in reality.
6.
Two similar triangles have sides in the ratio 2:5. If the perimeter of the smaller triangle is 18 cm, what is the perimeter of the larger triangle?
Answer: 45
1. What is the scale factor from smaller to larger? (as a decimal or fraction) 2.5
2. The perimeter of the smaller triangle is 18 cm. Multiply by the scale factor to find the larger perimeter. 45
7.
Two similar rectangles have lengths of 6 cm and 15 cm. If the width of the smaller rectangle is 4 cm, what is the width of the larger rectangle in cm?
Answer: 10
Scale factor = . Width of larger = cm.
8.
If , what is ?
1. a)5
2. b)15
3. c)9
4. d)45
Answer: 5
Cross-multiply: , so , therefore .
9.
A photo is 8 cm wide and 12 cm tall. It is enlarged so that the new width is 20 cm. What is the new height?
Answer: 30
1. What is the scale factor? (new width divided by old width) 2.5
2. Multiply the original height by the scale factor to get the new height. 30
10.
In two similar triangles, the ratio of corresponding sides is 3:7. If the shortest side of the larger triangle is 21 cm, what is the shortest side of the smaller triangle in cm?
Answer: 9
. Cross-multiply: , so cm.
11.
Triangle PQR is similar to triangle XYZ. In triangle PQR: PQ = 5 cm, QR = 8 cm, PR = 6 cm. In triangle XYZ: XY = 10 cm. Find YZ.
Answer: 16
1. PQ corresponds to XY. Find the scale factor (XY divided by PQ). 2
2. QR corresponds to YZ. Multiply QR by the scale factor to find YZ. 16
12.
A flagpole casts a shadow 15 m long. At the same time, a 2 m tall person casts a shadow 3 m long. How tall is the flagpole?
Answer: 10
1. Set up a proportion: height/shadow = height/shadow. What is ? 0.67
2. Set up the equation: . Cross-multiply: 30
3. Solve for h: 10
13.
Two similar pentagons have a scale factor of 4. If the area of the smaller pentagon is 25 square cm, what is the area of the larger pentagon in square cm?
Answer: 400
When the scale factor is , the area scales by . Area of larger = square cm.
14.
Rectangles A and B are similar. Rectangle A is 3 cm by 5 cm. Which of the following could be the dimensions of Rectangle B?
1. a)4 cm by 6 cm
2. b)6 cm by 8 cm
3. c)9 cm by 15 cm
4. d)5 cm by 7 cm
Answer: 9 cm by 15 cm
Rectangle A has ratio . Only (both multiplied by 3). The other options: , cannot be simplified to , .
15.
An architect creates a model of a building using a scale of 1:200. If the model is 45 cm tall, how tall is the actual building in meters?
Answer: 90
1. The scale 1:200 means 1 cm on the model = 200 cm in reality. Multiply 45 by 200. 9000
2. Convert 9000 cm to meters (divide by 100). 90
16.
Two similar triangles have areas in the ratio 9:25. What is the ratio of their corresponding sides? (Enter as a simplified ratio like 3:5)
Answer: 3:5
Area ratio = . Taking the square root, side ratio = .