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

For a point $(x,y)$ dilated by a scale - factor $k$ with center of dilation $(a,b)$, the new point $(x',y')$ is given by $x'=a + k(x - a)$ and $y'=b + k(y - b)$. Let the vertices of the triangle be $(x_1,y_1),(x_2,y_2),(x_3,y_3)$. Here $a = 3,b = 2,k = 2$.

Step2: Apply dilation

For a vertex $(x,y)$ of the triangle, $x'=3+2(x - 3)=2x-3$ and $y'=2 + 2(y - 2)=2y-2$. Calculate the new vertices after dilation.

Step3: Recall translation rule

The translation rule for moving a point $(x,y)$ two units left and three units down is $(x,y)\to(x - 2,y - 3)$.

Step4: Apply translation

For the dilated vertices $(x',y')$, the final vertices $(x_f,y_f)$ after translation are $x_f=x'-2=(2x - 3)-2=2x-5$ and $y_f=y'-3=(2y - 2)-3=2y-5$. Calculate the final coordinates of all vertices of the triangle.

Answer:

The new triangle is obtained by first dilating the original triangle's vertices using the dilation formula with center $(3,2)$ and scale - factor 2, and then translating the dilated vertices two units left and three units down using the translation rule $(x,y)\to(x - 2,y - 3)$. The specific coordinates of the new triangle's vertices depend on the original coordinates of the triangle's vertices. If the original vertices are $(x_1,y_1),(x_2,y_2),(x_3,y_3)$, the final vertices are $(2x_1-5,2y_1 - 5),(2x_2-5,2y_2 - 5),(2x_3-5,2y_3 - 5)$.