Questions: Here are two similar figures: scale 1 factor S2/s1=3 A2/A1=9 What are the scale factor and the area factor for this pair? Does the relationship make sense to you?

Here are two similar figures: scale 1 factor S2/s1=3 A2/A1=9 What are the scale factor and the area factor for this pair? Does the relationship make sense to you?
Transcript text: Here are two similar figures: scale 1 factor $\frac{S_{2}}{s_{1}}=3$ $\frac{A_{2}}{A_{1}}=9$ What are the scale factor and the area factor for this pair? Does the relationship make sense to you?
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Solution

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Solution Steps

Step 1: Find the scale factor

The larger triangle is twice the width and twice the height of the smaller triangle. Therefore, the scale factor is 2.

Step 2: Find the area factor

The area of the larger triangle can be calculated by drawing a rectangle around it, then subtracting the areas of the right triangles that form from the rectangle being divided. The area of the larger triangle is $(4_4 - \frac{1}{2}(4_2) - \frac{1}{2}(2_4) - \frac{1}{2}(2_2)) = 16-4-4-2=6$.

The area of the smaller triangle is $(2_2 - \frac{1}{2}(2_1)-\frac{1}{2}(1_2)-\frac{1}{2}(1_1))=4-1-1-\frac{1}{2}=\frac{3}{2}$. The ratio of the area of the larger triangle to the area of the smaller triangle is $\frac{6}{\frac{3}{2}} = 4$.

Therefore, the area factor is 4.

Step 3: Describe the relationship between the factors

The area factor is the square of the scale factor ($2^2=4$). This relationship is typical for similar figures.

Final Answer:

The scale factor is 2 and the area factor is 4. The area factor is the square of the scale factor, which makes sense.

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