1. **Problem Statement:** Find the scale factor of the enlargement from triangle PQR to P'Q'R', explain why it is negative, find the length of P'Q', the area of triangle PQR, and the area of triangle P'Q'R'. 2. **Given Data:** - P(2,2), Q(8,1), R(3,4) - P'(6,8), Q'(10,3), R'(7,5) - Center of enlargement C(5,2) - Length of PQ = $\sqrt{13}$ units 3. **Scale Factor Calculation:** The scale factor $k$ is the ratio of the distance from the center to a vertex in the image over the distance from the center to the corresponding vertex in the original. Calculate $CP$ and $CP'$: $$CP = \sqrt{(2-5)^2 + (2-2)^2} = \sqrt{(-3)^2 + 0^2} = 3$$ $$CP' = \sqrt{(6-5)^2 + (8-2)^2} = \sqrt{1^2 + 6^2} = \sqrt{1 + 36} = \sqrt{37}$$ Scale factor: $$k = \frac{CP'}{CP} = \frac{\sqrt{37}}{3}$$ 4. **Why is the scale factor negative?** The scale factor is negative because the image is on the opposite side of the center compared to the original point, indicating an enlargement with reflection. 5. **Length of P'Q':** Given $PQ = \sqrt{13}$, length scales by $|k|$: $$P'Q' = |k| \times PQ = \frac{\sqrt{37}}{3} \times \sqrt{13} = \frac{\sqrt{37 \times 13}}{3} = \frac{\sqrt{481}}{3}$$ 6. **Area of triangle PQR:** Use the coordinate formula for area: $$\text{Area} = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$$ Substitute: $$= \frac{1}{2} |2(1 - 4) + 8(4 - 2) + 3(2 - 1)|$$ $$= \frac{1}{2} |-6 + 16 + 3| = \frac{1}{2} |13| = 6.5$$ 7. **Area of triangle P'Q'R':** Area scales by $k^2$: $$\text{Area}_{P'Q'R'} = k^2 \times 6.5 = \left(\frac{\sqrt{37}}{3}\right)^2 \times 6.5 = \frac{37}{9} \times 6.5 = \frac{37 \times 6.5}{9} = \frac{240.5}{9} = 26.72$$ **Final answers:** - Scale factor size: $\frac{\sqrt{37}}{3}$ - Scale factor is negative because the image is on the opposite side of the center (enlargement with reflection). - Length of P'Q': $\frac{\sqrt{481}}{3}$ units - Area of triangle PQR: 6.5 square units - Area of triangle P'Q'R': 26.72 square units