Introduction to Similarity

Show your work for each problem.

1.
Two figures are similar if they have:
1. a)Different shapes and different sizes
2. b)Both the same shape and the same size
3. c)The same shape but not necessarily the same size
4. d)The same size but not necessarily the same shape
Answer: The same shape but not necessarily the same size
Similar figures have the same shape but can be different sizes. This is the key difference from congruent figures, which must be exactly the same shape AND size.
2.
Which symbol is used to show that two figures are similar?
1. a)
2. b)
3. c)
4. d)
Answer:
The symbol is used for similarity. The symbol is used for congruence, for equality, and for approximation.
3.
If the scale factor from figure A to figure B is , and a side of figure A is cm, what is the length of the corresponding side of figure B (in cm)?
Answer: 12
To find the corresponding side in the larger figure, multiply by the scale factor: cm.
4.
Are all squares similar to each other?
1. a)Only if they are rotated the same way
2. b)No, because squares can be different sizes
3. c)Yes, because all squares have the same angles and proportional sides
4. d)Only if they are the same size
Answer: Yes, because all squares have the same angles and proportional sides
Yes! All squares have four angles and four equal sides. The sides are always proportional because multiplying each side by the same scale factor gives another square.
5.
Triangle Triangle . If cm and cm, what is the scale factor from to ? (Enter as a decimal)
Answer: 1.5
The scale factor from to is . This means triangle is 1.5 times larger than triangle .
6.
Two rectangles have sides of cm by cm and cm by cm. Are they similar?
1. a)No, because the ratios are not equal ()
2. b)Yes, because both have right angles
3. c)No, because they are different sizes
4. d)Yes, because both are rectangles
Answer: No, because the ratios are not equal ()
For similar rectangles, all corresponding side ratios must be equal. Here, but . Since , the rectangles are NOT similar.
7.
Triangle Triangle . In : cm, cm, cm. In : cm. Find and .
Answer: YZ = 14 cm, XZ = 16 cm
1. Which sides correspond? corresponds to . What is the scale factor? 2
2. corresponds to . Using scale factor 2, what is ? 14
3. corresponds to . Using scale factor 2, what is ? 16
8.
On a map with scale 1:50000, two cities are 6 cm apart. What is the actual distance between them in kilometers?
Answer: 3
Distance = cm = km.
9.
An architect creates a 1:100 scale model of a building. The model is 45 cm tall and 30 cm wide. Find the actual building's height and width in meters.
Answer: Height = 45 m, Width = 30 m
1. What does scale 1:100 mean? 1 cm = 100 cm
2. The model is 45 cm tall. What is the real height in cm? 4500
3. Convert 4500 cm to meters. 45
4. The model is 30 cm wide. What is the real width in meters? 30
10.
In similar triangles and , cm, cm, and cm. Find (in cm).
Answer: 10
Using the proportion : . Cross multiply: . So cm.
11.
Triangle has angles , , and . Triangle has angles , , and an unknown angle. Are they similar? Find the unknown angle.
Answer: 60°, Yes they are similar
1. What is the sum of angles in any triangle? 180
2. In , we know and . What is the third angle? 60
3. Compare all three angles of both triangles. Are they the same? yes
4. If all angles match, are the triangles similar? yes
12.
If the scale factor from triangle A to triangle B is , what can we conclude?
1. a)Triangle B is smaller than triangle A
2. b)The triangles are not similar
3. c)The triangles are congruent
4. d)Triangle B is larger than triangle A
Answer: Triangle B is smaller than triangle A
When the scale factor is (less than 1), it means triangle B's sides are the length of triangle A's sides. Therefore, triangle B is smaller.
13.
A flagpole casts a shadow of 12 meters. At the same time, a 2-meter stick casts a shadow of 3 meters. How tall is the flagpole?
Answer: 8 meters
1. The flagpole and its shadow form a similar triangle to the stick and its shadow. Set up the proportion: height/12 = 2/3
2. Cross multiply to solve: 3h = 24
3. Solve for 8
14.
If with a scale factor of , and side cm, what is side (in cm)?
Answer: 12
With a scale factor of , each side of is 4 times the corresponding side of . So cm.
15.
Which of the following pairs of shapes are ALWAYS similar to each other?
1. a)Two circles
2. b)Two rectangles
3. c)Two triangles
4. d)Two parallelograms
Answer: Two circles
All circles are similar because they have no angles and their 'sides' (circumference) are all proportional to their diameter. Rectangles, triangles, and parallelograms can have different proportions, so they are not always similar.