1. **State the problem:** We have two triangles, GHI and G'H'I', where G'H'I' is a dilation of GHI. We need to find the scale factor of the dilation. 2. **Recall the formula for scale factor:** The scale factor $k$ of a dilation is the ratio of the lengths of corresponding sides in the image and the preimage: $$k = \frac{\text{length of side in } G'H'I'}{\text{length of corresponding side in } GHI}$$ 3. **Identify corresponding points:** Given points are: - $H(-10, -5)$ and $H'(-2, -2)$ - $G(-10, -10)$ and $G'(-2, -2)$ - $I(-5, 0)$ and $I'(-1, 0)$ 4. **Calculate lengths of sides in original triangle GHI:** - Side $GH$: distance between $G(-10, -10)$ and $H(-10, -5)$ $$GH = \sqrt{(-10 + 10)^2 + (-5 + 10)^2} = \sqrt{0^2 + 5^2} = 5$$ - Side $HI$: distance between $H(-10, -5)$ and $I(-5, 0)$ $$HI = \sqrt{(-5 + 10)^2 + (0 + 5)^2} = \sqrt{5^2 + 5^2} = \sqrt{50} = 5\sqrt{2}$$ - Side $GI$: distance between $G(-10, -10)$ and $I(-5, 0)$ $$GI = \sqrt{(-5 + 10)^2 + (0 + 10)^2} = \sqrt{5^2 + 10^2} = \sqrt{125} = 5\sqrt{5}$$ 5. **Calculate lengths of sides in image triangle G'H'I':** - Side $G'H'$: distance between $G'(-2, -2)$ and $H'(-2, -2)$ $$G'H' = \sqrt{(-2 + 2)^2 + (-2 + 2)^2} = \sqrt{0 + 0} = 0$$ Since $G'$ and $H'$ are the same point, this suggests a possible typo or error in the points given. However, we can check other sides. - Side $H'I'$: distance between $H'(-2, -2)$ and $I'(-1, 0)$ $$H'I' = \sqrt{(-1 + 2)^2 + (0 + 2)^2} = \sqrt{1^2 + 2^2} = \sqrt{5}$$ - Side $G'I'$: distance between $G'(-2, -2)$ and $I'(-1, 0)$ $$G'I' = \sqrt{(-1 + 2)^2 + (0 + 2)^2} = \sqrt{1 + 4} = \sqrt{5}$$ 6. **Calculate scale factor using side $H'I'$ and $HI$:** $$k = \frac{H'I'}{HI} = \frac{\sqrt{5}}{5\sqrt{2}} = \frac{\sqrt{5}}{5\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{10}}{10}$$ 7. **Simplify the scale factor:** $$k = \frac{\sqrt{10}}{10}$$ This is the scale factor of the dilation from triangle GHI to G'H'I'. **Final answer:** $$\boxed{\frac{\sqrt{10}}{10}}$$