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 \(K\) are \((- 8,8)\), the coordinates of point \(L\) are \((8,8)\), and the coordinates of point \(M\) are \((-4,6)\).

1. Then, use the dilation - formula for a dilation centered at the origin with a scale factor \(k\). The formula for a dilation centered at the origin with scale factor \(k\) is \((x,y)\to(kx,ky)\). Here, \(k = \frac{1}{2}\).

• For point \(K(-8,8)\):
• Calculate the new \(x\) - coordinate: \(x_{new}=k\times x=\frac{1}{2}\times(-8)=-4\).
• Calculate the new \(y\) - coordinate: \(y_{new}=k\times y=\frac{1}{2}\times8 = 4\). So the new coordinates of \(K\) are \((-4,4)\).
• For point \(L(8,8)\):
• Calculate the new \(x\) - coordinate: \(x_{new}=k\times x=\frac{1}{2}\times8 = 4\).
• Calculate the new \(y\) - coordinate: \(y_{new}=k\times y=\frac{1}{2}\times8 = 4\). So the new coordinates of \(L\) are \((4,4)\).
• For point \(M(-4,6)\):
• Calculate the new \(x\) - coordinate: \(x_{new}=k\times x=\frac{1}{2}\times(-4)=-2\).
• Calculate the new \(y\) - coordinate: \(y_{new}=k\times y=\frac{1}{2}\times6 = 3\). So the new coordinates of \(M\) are \((-2,3)\).

Answer:

The coordinates of \(K\) after dilation are \((-4,4)\), the coordinates of \(L\) after dilation are \((4,4)\), and the coordinates of \(M\) after dilation are \((-2,3)\).