Question

write the coordinates of the vertices after a dilation with a scale factor of $\frac{1}{3}$, centered at the origin.

Explanation:

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

• Assume the coordinates of point \(L\) are \((- 9,-9)\), the coordinates of point \(M\) are \((-9,9)\), and the coordinates of point \(N\) are \((9,-9)\).
• The rule for dilation centered at the origin with a scale - factor \(k\) is \((x,y)\to(kx,ky)\). Here, \(k = \frac{1}{3}\).

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

• For point \(L(-9,-9)\), when applying the dilation rule \((x,y)\to(kx,ky)\) with \(k=\frac{1}{3}\), we have \(x=-9\) and \(y = - 9\).
• The new \(x\) - coordinate is \(x_{new}=k\times x=\frac{1}{3}\times(-9)=-3\).
• The new \(y\) - coordinate is \(y_{new}=k\times y=\frac{1}{3}\times(-9)=-3\). So the new coordinates of \(L\) are \((-3,-3)\).

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

• For point \(M(-9,9)\), with \(x=-9\) and \(y = 9\) and \(k=\frac{1}{3}\).
• The new \(x\) - coordinate is \(x_{new}=k\times x=\frac{1}{3}\times(-9)=-3\).
• The new \(y\) - coordinate is \(y_{new}=k\times y=\frac{1}{3}\times9 = 3\). So the new coordinates of \(M\) are \((-3,3)\).

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

• For point \(N(9,-9)\), with \(x = 9\) and \(y=-9\) and \(k=\frac{1}{3}\).
• The new \(x\) - coordinate is \(x_{new}=k\times x=\frac{1}{3}\times9 = 3\).
• The new \(y\) - coordinate is \(y_{new}=k\times y=\frac{1}{3}\times(-9)=-3\). So the new coordinates of \(N\) are \((3,-3)\).

Step1: Identify dilation rule

The rule for dilation centered at the origin with scale - factor \(k=\frac{1}{3}\) is \((x,y)\to(\frac{1}{3}x,\frac{1}{3}y)\).

Step2: Calculate new coordinates of \(L\)

Given \(L(-9,-9)\), new \(x=\frac{1}{3}\times(-9)=-3\), new \(y=\frac{1}{3}\times(-9)=-3\).

Step3: Calculate new coordinates of \(M\)

Given \(M(-9,9)\), new \(x=\frac{1}{3}\times(-9)=-3\), new \(y=\frac{1}{3}\times9 = 3\).

Step4: Calculate new coordinates of \(N\)

Given \(N(9,-9)\), new \(x=\frac{1}{3}\times9 = 3\), new \(y=\frac{1}{3}\times(-9)=-3\).

Answer:

The new coordinates of \(L\) are \((-3,-3)\), the new coordinates of \(M\) are \((-3,3)\), and the new coordinates of \(N\) are \((3,-3)\).