Dilations & Similarity - Pre Algebra

1. Two similar triangles have areas in the ratio $16 : 25$. What scale factor maps the smaller onto the larger?

   $\frac{5}{4}$. The side ratio is the square root of the area ratio: $\frac{\sqrt{25}}{\sqrt{16}} = \frac{5}{4}$.

2. When a figure is dilated by scale factor $k$, by what factor is its area multiplied?

   $k^2$, because both dimensions are multiplied by $k$.

3. What does a dilation scale factor greater than 1 do, and what does one between 0 and 1 do?

   Greater than 1 enlarges the figure; between 0 and 1 shrinks it.

4. What is the image of $(2, 5)$ after a dilation with scale factor 3 centered at the origin?

   $(6, 15)$. Multiply each coordinate by 3.

5. How are similar figures defined using transformations?

   Two figures are similar if one can be mapped onto the other by a sequence of rigid motions and dilations.

6. A side of length 5 has an image of length 15 after a dilation. What is the scale factor?

   3, since $\frac{15}{5} = 3$.

7. A segment 8 units long is dilated by scale factor $\frac{3}{4}$. How long is its image?

   6 units, since $8 \times \frac{3}{4} = 6$.

8. What is the image of $(6, -4)$ after a dilation with scale factor $\frac{1}{2}$ centered at the origin?

   $(3, -2)$. Multiply each coordinate by $\frac{1}{2}$.

9. What is the image of a figure under a dilation with scale factor 1?

   The figure itself: every point maps to itself, so the image is congruent to the original.

10. A figure with area 32 square units is dilated by scale factor $\frac{3}{4}$. What is the area of the image?

    18 square units, since $32 \times \frac{9}{16} = 18$.

11. A triangle with perimeter 28 units is dilated by scale factor $\frac{3}{4}$. What is the perimeter of the image?

    21 units. Perimeter is multiplied by the scale factor: $28 \times \frac{3}{4} = 21$.

12. Which measures does a dilation keep unchanged?

    Angle measures. Side lengths are all multiplied by the scale factor, so the shape is kept.

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