Directions: Find the centroid of each triangle. \( \begin{array}{l}\text { 1. } \mathrm{A}(3,-2), \mathrm{B}(0,6) \text { and } \mathrm{C}(-3,-2) \\ \text { 3. } \mathrm{U}(-1,-5), \mathrm{V}(-4,0) \text { and } \mathrm{W}(-2,-3) \\ \text { 5. } \mathrm{P}(1,-4), \mathrm{Q}(0,5) \text { and } \mathrm{R}(-9,10) \\ \text { 7. } \mathrm{P}(6,6), \mathrm{Q}(1,4) \text { and } \mathrm{R}(5,2) \\ \text { 4. } \mathrm{X}(-1,0), \mathrm{Y}(3,5) \text { and } \mathrm{Z}(4,-2) \\ \text { 6. } \mathrm{A}(2,3), \mathrm{B}(1,7) \text { and } \mathrm{C}(4,5) \\ \text { 8. } \mathrm{S}(7,8), \mathrm{T}(-4,3) \text { and } \mathrm{U}(0,10)\end{array} \)

To find the centroid of a triangle given its vertices, we can use the formula for the centroid \( G \) of a triangle with vertices \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \): \[ G\left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \] Now, let's calculate the centroids for each triangle step by step. ### 1. Triangle A(3,-2), B(0,6), C(-3,-2) - \( x_1 = 3, y_1 = -2 \) - \( x_2 = 0, y_2 = 6 \) - \( x_3 = -3, y_3 = -2 \) Calculating the centroid: \[ G_x = \frac{3 + 0 - 3}{3} = \frac{0}{3} = 0 \] \[ G_y = \frac{-2 + 6 - 2}{3} = \frac{2}{3} \] **Centroid**: \( G(0, \frac{2}{3}) \) ### 2. Triangle U(-1,-5), V(-4,0), W(-2,-3) - \( x_1 = -1, y_1 = -5 \) - \( x_2 = -4, y_2 = 0 \) - \( x_3 = -2, y_3 = -3 \) Calculating the centroid: \[ G_x = \frac{-1 - 4 - 2}{3} = \frac{-7}{3} \] \[ G_y = \frac{-5 + 0 - 3}{3} = \frac{-8}{3} \] **Centroid**: \( G\left(-\frac{7}{3}, -\frac{8}{3}\right) \) ### 3. Triangle P(1,-4), Q(0,5), R(-9,10) - \( x_1 = 1, y_1 = -4 \) - \( x_2 = 0, y_2 = 5 \) - \( x_3 = -9, y_3 = 10 \) Calculating the centroid: \[ G_x = \frac{1 + 0 - 9}{3} = \frac{-8}{3} \] \[ G_y = \frac{-4 + 5 + 10}{3} = \frac{11}{3} \] **Centroid**: \( G\left(-\frac{8}{3}, \frac{11}{3}\right) \) ### 4. Triangle P(6,6), Q(1,4), R(5,2) - \( x_1 = 6, y_1 = 6 \) - \( x_2 = 1, y_2 = 4 \) - \( x_3 = 5, y_3 = 2 \) Calculating the centroid: \[ G_x = \frac{6 + 1 + 5}{3} = \frac{12}{3} = 4 \] \[ G_y = \frac{6 + 4 + 2}{3} = \frac{12}{3} = 4 \] **Centroid**: \( G(4, 4) \) ### 5. Triangle X(-1,0), Y(3,5), Z(4,-2) - \( x_1 = -1, y_1 = 0 \) - \( x_2 = 3, y_2 = 5 \) - \( x_3 = 4, y_3 = -2 \) Calculating the centroid: \[ G_x = \frac{-1 + 3 + 4}{3} = \frac{6}{3} = 2 \] \[ G_y = \frac{0 + 5 - 2}{3} = \frac{3}{3} = 1 \] **Centroid**: \( G(2, 1) \) ### 6. Triangle A(2,3), B(1,7), C(4,5) - \( x_1 = 2, y_1 = 3 \) - \( x_2 = 1, y_2 = 7 \) - \( x_3 = 4, y_3 = 5 \) Calculating the centroid: \[ G_x = \frac{2 + 1 + 4}{3} = \frac{7}{3} \] \[ G_y = \frac{3 + 7 + 5}{3} = \frac{15}{3} = 5 \] **Centroid**: \( G\left(\frac{7}{3}, 5\right) \) ### 7. Triangle S(7,8), T(-4,3), U(0,10) - \( x_1 = 7, y_1 = 8 \) - \( x_2 = -4, y_2 = 3 \) - \( x_3 = 0, y_3 = 10 \) Calculating the centroid: \[ G_x = \frac{7 - 4 + 0}{3} = \frac{3}{3} = 1 \] \[ G_y = \frac{8 + 3 + 10}{3} = \frac{21}{3} = 7 \] **Centroid**: \( G(1, 7) \) ### Summary of Centroids 1. \( G(0, \frac{2}{3}) \) 2. \( G\left(-\frac{7}{3}, -\frac{8}{3}\right) \) 3. \( G\left(-\frac{8}{3}, \frac{11}{3}\right) \) 4. \( G(4, 4) \) 5. \( G(2, 1) \) 6. \( G\left(\frac{7}{3}, 5\right) \) 7. \( G(1, 7) \)