Question

write the coordinates of the vertices after a dilation with a scale factor of $\frac{1}{2}$, centered at the origin.

Explanation:

1. First, identify the original coordinates of the vertices:

• From the graph, the coordinates of point \(A\) are \((- 4,-6)\), the coordinates of point \(B\) are \((0,-6)\), and the coordinates of point \(C\) are \((0, - 10)\).

1. Recall the rule for dilation centered at the origin:

• If a point \((x,y)\) is dilated with a scale - factor \(k\) centered at the origin, the new coordinates \((x',y')\) are given by \((x',y')=(k x,k y)\). Here, \(k = \frac{1}{2}\).

1. Calculate the new coordinates of point \(A\):

• For point \(A(-4,-6)\), \(x=-4\) and \(y = - 6\).
• \(x'=\frac{1}{2}\times(-4)=-2\) and \(y'=\frac{1}{2}\times(-6)=-3\). So the new coordinates of \(A\) are \((-2,-3)\).

1. Calculate the new coordinates of point \(B\):

• For point \(B(0,-6)\), \(x = 0\) and \(y=-6\).
• \(x'=\frac{1}{2}\times0 = 0\) and \(y'=\frac{1}{2}\times(-6)=-3\). So the new coordinates of \(B\) are \((0,-3)\).

1. Calculate the new coordinates of point \(C\):

• For point \(C(0,-10)\), \(x = 0\) and \(y=-10\).
• \(x'=\frac{1}{2}\times0 = 0\) and \(y'=\frac{1}{2}\times(-10)=-5\). So the new coordinates of \(C\) are \((0,-5)\).

Step1: Identify original coordinates

\(A(-4,-6)\), \(B(0,-6)\), \(C(0,-10)\)

Step2: Apply dilation formula for \(A\)

\(x'=\frac{1}{2}\times(-4),y'=\frac{1}{2}\times(-6)\)

Step3: Apply dilation formula for \(B\)

\(x'=\frac{1}{2}\times0,y'=\frac{1}{2}\times(-6)\)

Step4: Apply dilation formula for \(C\)

\(x'=\frac{1}{2}\times0,y'=\frac{1}{2}\times(-10)\)

Answer:

The new coordinates of \(A\) are \((-2,-3)\), the new coordinates of \(B\) are \((0,-3)\), and the new coordinates of \(C\) are \((0,-5)\)