Question

write the coordinates of the vertices after a dilation with a scale factor of $\frac{1}{5}$, centered at the origin.

Explanation:

1. First, assume the coordinates of the vertices of the original figure:

• Let's assume the vertices of the original figure are \(O(0,0)\), \(Q(10, - 10)\), and \(R(10,5)\).
• The formula for dilation centered at the origin with a scale - factor \(k\) is \((x,y)\to(kx,ky)\). Here, \(k = \frac{1}{5}\).

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

• For point \(O(0,0)\), when we apply the dilation formula \((x,y)\to(kx,ky)\) with \(k=\frac{1}{5}\), we have \(x = 0\) and \(y = 0\). Then \(kx=\frac{1}{5}\times0 = 0\) and \(ky=\frac{1}{5}\times0 = 0\). So the new coordinates of \(O\) are \((0,0)\).

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

• For point \(Q(10,-10)\), using the dilation formula \((x,y)\to(kx,ky)\) with \(k = \frac{1}{5}\), we get \(kx=\frac{1}{5}\times10 = 2\) and \(ky=\frac{1}{5}\times(-10)=-2\). So the new coordinates of \(Q\) are \((2,-2)\).

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

• For point \(R(10,5)\), using the dilation formula \((x,y)\to(kx,ky)\) with \(k=\frac{1}{5}\), we have \(kx=\frac{1}{5}\times10 = 2\) and \(ky=\frac{1}{5}\times5 = 1\). So the new coordinates of \(R\) are \((2,1)\).

Answer:

The coordinates of the vertices after dilation are \(O(0,0)\), \(Q(2,-2)\), \(R(2,1)\)