Question

a. draw a scaled copy of the polygon using a scale factor 3. label the copy a.
b. draw a scaled copy of the polygon with a scale factor \\(\frac{1}{2}\\). label it b.
c. is polygon a a scaled copy of polygon b? if so, what is the scale factor that takes b to a?
(from unit 1, lesson 3)

Explanation:

Part a and b (Drawing Instructions)

To draw scaled copies:

1. Identify the original polygon's dimensions: First, analyze the original polygon on the grid. Let's assume the original polygon has a rectangular part (e.g., width 2 units, height 2 units) and a triangular part (base 2 units, height 2 units, etc.—exact dimensions depend on grid squares, but the key is to scale each side by the given factor).
2. Scale for factor 3 (Polygon A): For each side length of the original polygon, multiply by 3. For example, if a side is 2 grid units, the scaled side becomes \( 2\times3 = 6 \) grid units. Draw the new polygon with these scaled side lengths, maintaining the same shape (proportions).
3. Scale for factor \( \frac{1}{2} \) (Polygon B): For each side length of the original polygon, multiply by \( \frac{1}{2} \). For example, if a side is 2 grid units, the scaled side becomes \( 2\times\frac{1}{2}=1 \) grid unit. Draw the new polygon with these scaled side lengths, maintaining the same shape.

Part c

Step1: Recall Scaled Copy Definition

A scaled copy has all sides scaled by the same factor, preserving shape (similar figures). Polygon A is scaled from original by 3, Polygon B by \( \frac{1}{2} \). So A and B are similar (same shape, different size).

Step2: Find Scale Factor from B to A

Let the original polygon have side length \( s \).

• Side length of A: \( s \times 3 \)
• Side length of B: \( s \times \frac{1}{2} \)

To find the scale factor \( k \) from B to A: \( \text{Side of A} = k \times \text{Side of B} \)
Substitute: \( 3s = k \times \frac{1}{2}s \)

Step3: Solve for \( k \)

Divide both sides by \( s \) ( \( s
eq 0 \) ): \( 3 = k \times \frac{1}{2} \)
Multiply both sides by 2: \( k = 3\times2 = 6 \)

Answer:

a. (Drawing: Scale each side of the original polygon by 3 and redraw, label as A.)
b. (Drawing: Scale each side of the original polygon by \( \frac{1}{2} \) and redraw, label as B.)
c. Yes, Polygon A is a scaled copy of Polygon B. The scale factor from B to A is \( \boldsymbol{6} \).