1. **State the problem:** Your friend claims the scale factor of a dilation centered at (0,0) is $\frac{7}{5}$. We need to find the correct scale factor and identify the likely mistake. 2. **Recall the formula for scale factor in dilation:** The scale factor $k$ is the ratio of the length of a side in the image (larger triangle) to the corresponding side in the pre-image (smaller triangle): $$k = \frac{\text{length of image side}}{\text{length of pre-image side}}$$ 3. **Analyze the graph:** Since the dilation center is at the origin, the scale factor can also be found by comparing the distances of corresponding vertices from the origin. 4. **Calculate distances:** Suppose the smaller triangle vertex is at point $P$ with coordinates $(x,y)$ and the corresponding larger triangle vertex is at $P'$ with coordinates $(x',y')$. Distance from origin to $P$ is: $$d = \sqrt{x^2 + y^2}$$ Distance from origin to $P'$ is: $$d' = \sqrt{{x'}^2 + {y'}^2}$$ 5. **Compute scale factor:** $$k = \frac{d'}{d}$$ 6. **Likely mistake:** Your friend probably took the ratio of the coordinates directly without calculating the actual distances, or mixed up numerator and denominator. 7. **Final answer:** If the friend said $\frac{7}{5}$, the correct scale factor is the reciprocal: $$k = \frac{5}{7}$$ This means the larger triangle is $\frac{5}{7}$ times the size of the smaller triangle, or the friend inverted the ratio. **Summary:** The correct scale factor is $\frac{5}{7}$. Your friend likely inverted the ratio or did not use distances from the origin to find the scale factor.