Question

5.) dilate the triangle by a scale factor of 2, with the center of dilation at a(3, 2) - translate the dilated triangle two units left and three units down

Explanation:

Step1: Recall dilation formula

Let the center of dilation be $A(x_0,y_0)=(3,2)$ and scale - factor $k = 2$. For a point $P(x,y)$ in the original triangle, the coordinates of the dilated point $P'(x',y')$ are given by $x'=x_0 + k(x - x_0)$ and $y'=y_0 + k(y - y_0)$.

Step2: Apply dilation

Suppose the vertices of the original triangle are $(x_1,y_1),(x_2,y_2),(x_3,y_3)$. After dilation, the new vertices $(x_1',y_1'),(x_2',y_2'),(x_3',y_3')$ are calculated as follows:
$x_i'=3 + 2(x_i - 3)=2x_i-3$ and $y_i'=2 + 2(y_i - 2)=2y_i - 2$ for $i = 1,2,3$.

Step3: Recall translation formula

The translation is 2 units left and 3 units down. The formula for translation is $x''=x'-2$ and $y''=y'-3$.

Step4: Apply translation

Substitute the dilated - point coordinates into the translation formula. For a dilated point $(x',y')$, the final - point coordinates $(x'',y'')$ are:
$x''=(2x_i-3)-2=2x_i - 5$ and $y''=(2y_i - 2)-3=2y_i-5$ for $i = 1,2,3$. Then plot the new - triangle with these final - point coordinates on the coordinate grid.

Since we don't have the original coordinates of the triangle vertices, we can't give numerical answers. But the general steps for transformation are as above. If we assume the original vertices of the triangle are $(x_1,y_1),(x_2,y_2),(x_3,y_3)$:

1. Dilate:

• $x_1'=2x_1 - 3,y_1'=2y_1 - 2$
• $x_2'=2x_2 - 3,y_2'=2y_2 - 2$
• $x_3'=2x_3 - 3,y_3'=2y_3 - 2$

1. Translate:

• $x_1''=2x_1-5,y_1''=2y_1 - 5$
• $x_2''=2x_2-5,y_2''=2y_2 - 5$
• $x_3''=2x_3-5,y_3''=2y_3 - 5$

Answer:

The new vertices of the triangle after dilation by a scale - factor of 2 with center of dilation at $(3,2)$ and then translation 2 units left and 3 units down are obtained by the formulas $x'' = 2x-5$ and $y''=2y - 5$ for each vertex $(x,y)$ of the original triangle.