Unit 1, Day 2: Introduce concept of scales (dilations)

Objective: Introduction on concept of scales and review from day 1.

Scale Factor

Definition of Scale Factor

• The ratio of any two corresponding lengths in two similar geometric figures is called as Scale Factor.
• The ratio of the length of the scale drawing to the corresponding length of the actual object is called as Scale Factor.

More about Scale Factor

• A scale factor is a number used as a multiplier in scaling.
• A scale factor is used to scale shapes in 1, 2, or 3 dimensions.
• Scale factor can be found in the following scenarios:
  1. Size Transformation: In size transformation, the scale factor is the ratio of expressing the amount of magnification.
  2. Scale Drawing: In scale drawing, the scale factor is the ratio of measurement of the drawing compared to the measurement of the original figure.
  3. Comparing Two Similar Geometric Figures: The scale factor when comparing two similar geometric figures, is the ratio of lengths of the corresponding sides.

ABCD and PQRS are similar polygons. Then the scale factor of polygon ABCD to polygon PQRS is the ratio of the lengths of the corresponding sides.

• Scale factor = BC:QR = 3:8.

Solved Example on Scale Factor

Find the scale factor from the larger rectangle to the smaller rectangle, if the two rectangles are similar.

Choices: 
A. 5:1
B. 5:6
C. 6:5
D. 6:7
Correct Answer: B
Solution:
Step 1: If we multiply the length of one side of the larger rectangle by the scale factor we get the length of the corresponding side of the smaller rectangle.
Step 2: Dimension of larger rectangle × scale factor = dimension of smaller rectangle
Step 3: 24 × scale factor = 20 [Substitute the values.]
Step 4: Scale factor = 20/24 [Divide each side by 24.]
Step 5: Scale factor =  = 5:6 [Simplify.]
Therefore, scale factor from the larger rectangle to the smaller rectangle is 5:6.

Questions: 

1. Find the image of point P (1, 2) under dilation with center (0, 0) with a scale factor of 2.

A. (- 1, - 2)

B. (2, 4)

C. (- 2, 4)

D. (2, - 4)

2. A map is constructed on a scale of 10 ft to 1 in. What is the area of the land represented on the map by an oblong measuring 4 × 5 in.?

A. 4000 ft2

B. 2000 ft2

C. 50 ft2

D. 2100 ft2

3. A map is constructed on a scale of 20 yd to 1 in. What is the area of the land represented on the map by an oblong measuring 3 × 2 in.?

A. 40 yd2

B. 2400 yd2

C. 4800 yd2

D. 2500 yd2 Answers: 1. Step 1 : To find the image of a point on the coordinate plane under dilation with center as origin, multiply the coordinates with scale factor. Step 2 : Image of P (1, 2) is P ′(1 × 2, 2 × 2) Step 3 : = (2, 4) 2.Step 1 : Scale = 10 ft to 1 in. Step 2 : The actual width of the ground = 4 × 10 = 40 ft. Step 3 : The actual length of the ground = 5 × 10 = 50 ft Step 4 : So, the area of the land = 40 × 50 = 2000 ft2. 3. Step 1 : Scale = 20 yd to 1 in. Step 2 : The actual width of the ground = 3 × 20 = 60 yd. Step 3 : The actual length of the ground = 2 × 20 = 40 yd. Step 4 : So, the area of the land = 60 × 40 = 2400 yd2.