Identify the coordinates of the original triangle and the center of enlargement.

Question

Identify the coordinates of the original triangle and the center of enlargement.

Asked on March 29, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-03-29T17:49:46.643Z

Step 1: Identify the coordinates of the original triangle and the center of enlargement. The original triangle has vertices $A(-1, -1)$, $B(-1, -4)$, and $C(-3, -2)$. The center of enlargement is $O(2, 1)$. The scale factor is $k=2$. Step 2: Calculate the coordinates of the enlarged triangle using the formula for enlargement. For a point $(x, y)$ enlarged from a center $(x_c, y_c)$ with a scale factor $k$, the new point $(x', y')$ is given by: $$x' = x_c + k(x - x_c)$$ $$y' = y_c + k(y - y_c)$$ For point $A(-1, -1)$: $$x_A' = 2 + 2(-1 - 2) = 2 + 2(-3) = 2 - 6 = -4$$ $$y_A' = 1 + 2(-1 - 1) = 1 + 2(-2) = 1 - 4 = -3$$ So, $A' = (-4, -3)$. For point $B(-1, -4)$: $$x_B' = 2 + 2(-1 - 2) = 2 + 2(-3) = 2 - 6 = -4$$ $$y_B' = 1 + 2(-4 - 1) = 1 + 2(-5) = 1 - 10 = -9$$ So, $B' = (-4, -9)$. For point $C(-3, -2)$: $$x_C' = 2 + 2(-3 - 2) = 2 + 2(-5) = 2 - 10 = -8$$ $$y_C' = 1 + 2(-2 - 1) = 1 + 2(-3) = 1 - 6 = -5$$ So, $C' = (-8, -5)$. The coordinates of the enlarged triangle are $A'(-4, -3)$, $B'(-4, -9)$, and $C'(-8, -5)$. Step 3: Draw the triangles on the Cartesian plane. • Draw the x-axis and y-axis on the grid provided. • Plot the center of enlargement $O(2, 1)$. • Plot the vertices of the original triangle: $A(-1, -1)$, $B(-1, -4)$, and $C(-3, -2)$. Connect these points to form triangle ABC. • Plot the vertices of the enlarged triangle: $A'(-4, -3)$, $B'(-4, -9)$, and $C'(-8, -5)$. Connect these points to form triangle A'B'C'. The coordinates of the original triangle are: $A = (-1, -1)$ $B = (-1, -4)$ $C = (-3, -2)$ The coordinates of the enlarged triangle are: $A' = (-4, -3)$ $B' = (-4, -9)$ $C' = (-8, -5)$ The center of enlargement is $O = (2, 1)$.

Accepted Answer · 2026-03-29T17:49:46.643Z

Step 1: Identify the coordinates of the original triangle and the center of enlargement. The original triangle has vertices $A(-1, -1)$, $B(-1, -4)$, and $C(-3, -2)$. The center of enlargement is $O(2, 1)$. The scale factor is $k=2$. Step 2: Calculate the coordinates of the enlarged triangle using the formula for enlargement. For a point $(x, y)$ enlarged from a center $(x_c, y_c)$ with a scale factor $k$, the new point $(x', y')$ is given by: $$x' = x_c + k(x - x_c)$$ $$y' = y_c + k(y - y_c)$$ For point $A(-1, -1)$: $$x_A' = 2 + 2(-1 - 2) = 2 + 2(-3) = 2 - 6 = -4$$ $$y_A' = 1 + 2(-1 - 1) = 1 + 2(-2) = 1 - 4 = -3$$ So, $A' = (-4, -3)$. For point $B(-1, -4)$: $$x_B' = 2 + 2(-1 - 2) = 2 + 2(-3) = 2 - 6 = -4$$ $$y_B' = 1 + 2(-4 - 1) = 1 + 2(-5) = 1 - 10 = -9$$ So, $B' = (-4, -9)$. For point $C(-3, -2)$: $$x_C' = 2 + 2(-3 - 2) = 2 + 2(-5) = 2 - 10 = -8$$ $$y_C' = 1 + 2(-2 - 1) = 1 + 2(-3) = 1 - 6 = -5$$ So, $C' = (-8, -5)$. The coordinates of the enlarged triangle are $A'(-4, -3)$, $B'(-4, -9)$, and $C'(-8, -5)$. Step 3: Draw the triangles on the Cartesian plane. • Draw the x-axis and y-axis on the grid provided. • Plot the center of enlargement $O(2, 1)$. • Plot the vertices of the original triangle: $A(-1, -1)$, $B(-1, -4)$, and $C(-3, -2)$. Connect these points to form triangle ABC. • Plot the vertices of the enlarged triangle: $A'(-4, -3)$, $B'(-4, -9)$, and $C'(-8, -5)$. Connect these points to form triangle A'B'C'. The coordinates of the original triangle are: $A = (-1, -1)$ $B = (-1, -4)$ $C = (-3, -2)$ The coordinates of the enlarged triangle are: $A' = (-4, -3)$ $B' = (-4, -9)$ $C' = (-8, -5)$ The center of enlargement is $O = (2, 1)$.