Question

△rst is dilated by a scale factor of 5 with a center of dilation at the origin to form △rst. then △rst is dilated by a scale factor of \frac{2}{5} with a center of dilation at the origin to form △rst. which statement is true? r has coordinates (15, -10). r has coordinates (-4,6). t has coordinates (2, -4). s has coordinates (-20, -15).

Explanation:

Step1: Find coordinates of original points

From the graph, assume \(R(3, - 2)\), \(S(-2,4)\), \(T(-2,-1)\)

Step2: First dilation (scale factor \(5\))

The formula for dilation about the origin \((x,y)\to(5x,5y)\)
For \(R(3,-2)\), \(R'(3\times5,-2\times5)=(15,-10)\)
For \(S(-2,4)\), \(S'(-2\times5,4\times5)=(-10,20)\)
For \(T(-2,-1)\), \(T'(-2\times5,-1\times5)=(-10,-5)\)

Step3: Second dilation (scale factor \(\frac{2}{5}\))

The formula for dilation about the origin \((x,y)\to(\frac{2}{5}x,\frac{2}{5}y)\)
For \(R'(15,-10)\), \(R''(15\times\frac{2}{5},-10\times\frac{2}{5})=(6,-4)\)
For \(S'(-10,20)\), \(S''(-10\times\frac{2}{5},20\times\frac{2}{5})=(-4,8)\)
For \(T'(-10,-5)\), \(T''(-10\times\frac{2}{5},-5\times\frac{2}{5})=(-4,-2)\)

Answer:

\(R'\) has coordinates \((15, - 10)\)