Question

draw the image of the triangle under a dilation with the center at the black point and a scale factor of 2.

Explanation:

Step1: Find the vector from center to vertices

Let the center (black point) be \(C(x_0,y_0)\) and a vertex of the triangle be \(V(x_1,y_1)\). The vector \(\vec{CV}=(x_1 - x_0,y_1 - y_0)\)

Step2: Apply dilation

For a scale factor \(k = 2\), the new vertex \(V'\) has coordinates \((x_0+2(x_1 - x_0),y_0+2(y_1 - y_0))=(2x_1 - x_0,2y_1 - y_0)\)

Step3: Repeat for all vertices

Do the above calculation for each of the three vertices of the triangle. Then connect the new vertices to form the dilated triangle.

Answer:

Draw the new triangle with vertices obtained by the dilation formula \((2x_1 - x_0,2y_1 - y_0)\) for each original vertex \((x_1,y_1)\) with center \((x_0,y_0)\) (the black - point)