Question

6-116. samantha is dilating triangle ( abc ) at right. she multiplied each ( x )-coordinate and ( y )-coordinate of triangle ( abc ) by ( -2 ).

a. graph samantha’s new triangle.

b. describe how triangle ( abc ) changed.

Explanation:

First, identify the coordinates of triangle \(ABC\) from the graph:

• \(A = (0, 3)\)
• \(B = (-2, 0)\)
• \(C = (2, 1)\)

Step1: Find new coordinates

Multiply each \(x,y\) by \(-2\):
\(A' = (0 \times -2, 3 \times -2) = (0, -6)\)
\(B' = (-2 \times -2, 0 \times -2) = (4, 0)\)
\(C' = (2 \times -2, 1 \times -2) = (-4, -2)\)

Step2: Graph the new triangle

Plot \(A'(0,-6)\), \(B'(4,0)\), \(C'(-4,-2)\) and connect the points.

Step3: Describe the transformation

Analyze scale and orientation change.

Answer:

a. To graph the new triangle \(A'B'C'\), plot the points \(A'(0, -6)\), \(B'(4, 0)\), and \(C'(-4, -2)\), then connect the vertices with line segments.

b. Triangle \(ABC\) was dilated by a scale factor of 2 (so all side lengths doubled) and reflected across both the \(x\)-axis and \(y\)-axis (equivalent to a reflection through the origin) to form the new triangle \(A'B'C'\).