Question

write the coordinates of the vertices after a dilation with a scale factor of 1/3, centered at the origin.

Explanation:

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

• Assume the coordinates of point \(A\) are \((- 10,10)\), point \(C\) are \((10,10)\), and point \(D\) are \((-10, - 10)\) (from the grid - the origin is \((0,0)\)).
• The formula for dilation centered at the origin with a scale factor \(k\) is \((x,y)\to(kx,ky)\). Here, \(k = \frac{1}{2}\).

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

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

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

• For point \(C(10,10)\), with \(x = 10\) and \(y = 10\) and \(k=\frac{1}{2}\).
• The new \(x\) - coordinate is \(k\times x=\frac{1}{2}\times10 = 5\).
• The new \(y\) - coordinate is \(k\times y=\frac{1}{2}\times10 = 5\). So the new coordinates of \(C\) are \((5,5)\).

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

• For point \(D(-10,-10)\), with \(x=-10\) and \(y=-10\) and \(k = \frac{1}{2}\).
• The new \(x\) - coordinate is \(k\times x=\frac{1}{2}\times(-10)=-5\).
• The new \(y\) - coordinate is \(k\times y=\frac{1}{2}\times(-10)=-5\). So the new coordinates of \(D\) are \((-5,-5)\).

Step1: Identify original coordinates

Assume \(A(-10,10)\), \(C(10,10)\), \(D(-10,-10)\)

Step2: Apply dilation formula for \(A\)

\((-10,10)\to(\frac{1}{2}\times(-10),\frac{1}{2}\times10)=(-5,5)\)

Step3: Apply dilation formula for \(C\)

\((10,10)\to(\frac{1}{2}\times10,\frac{1}{2}\times10)=(5,5)\)

Step4: Apply dilation formula for \(D\)

\((-10,-10)\to(\frac{1}{2}\times(-10),\frac{1}{2}\times(-10))=(-5,-5)\)

Answer:

The new coordinates of \(A\) are \((-5,5)\), of \(C\) are \((5,5)\), and of \(D\) are \((-5,-5)\)