Scale Factor Effects - Pre Algebra

1. A triangle with area 48 square feet is scaled by a factor of $\frac{1}{2}$. What is the new area?

   12 square feet. Area scales by $\left(\frac{1}{2}\right)^2 = \frac{1}{4}$: $48 \times \frac{1}{4} = 12$.

2. Every side of a figure with area 5 square feet is doubled. What is the new area?

   20 square feet. Doubling lengths multiplies area by $2^2 = 4$.

3. A figure is scaled by a factor of $k$. By what factor is its perimeter multiplied?

   $k$. Every length, including the perimeter, is multiplied by the scale factor.

4. A figure is enlarged by a scale factor of 4. What is the ratio of the new area to the original area?

   $16:1$. Area scales by $k^2 = 4^2 = 16$.

5. A square with perimeter 12 cm is scaled by a factor of 3. What is the new perimeter?

   36 cm. Perimeter scales by $k$: $12 \times 3 = 36$.

6. A rectangle with area 10 square inches is scaled by a factor of 3. What is the new area?

   90 square inches. Area scales by $k^2 = 9$: $10 \times 9 = 90$.

7. Two similar figures have areas 9 and 36 square units. What is the scale factor from the smaller to the larger?

   $2$. The area ratio is $\frac{36}{9} = 4 = k^2$, so $k = \sqrt{4} = 2$.

8. A student says doubling the sides of a square doubles its area. What is the error?

   Area is multiplied by $2^2 = 4$, not 2. For example, a $1 \times 1$ square (area 1) becomes $2 \times 2$ (area 4).

9. A figure is scaled by a factor of $k$. By what factor is its area multiplied?

   $k^2$. Area depends on two lengths, and each one is multiplied by $k$.

10. A figure with perimeter 18 m is scaled by a factor of $\frac{1}{3}$. What is the new perimeter?

    6 m. Perimeter scales by $k$: $18 \times \frac{1}{3} = 6$.

11. Two similar polygons have perimeters 10 and 25. What is the ratio of their areas?

    $4:25$. The scale factor is $10:25 = 2:5$, and areas scale by its square.

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