Question

what translation and dilation will map
△qrs onto its image?
△qrs is translated? units to the
right, then dilated by a scale factor of
? with a center of dilation at
the origin.

Explanation:

Step1: Find the translation

Looking at the \(x -\)coordinates of corresponding points (e.g., \(Q(- 4,1)\) and \(Q''(4,4)\)). The \(x -\)coordinate of \(Q\) is \(-4\) and of \(Q''\) is \(4\). The translation in the \(x -\)direction (horizontal translation) is calculated as \(4-(-4)=8\) units to the right.

Step2: Find the scale factor

Take a side length. For example, the length of \(QR\) in \(\triangle QRS\): \(QR\) has endpoints \(Q(-4,1)\) and \(R(-2,1)\), so \(QR=\vert-2 - (-4)\vert=2\).
The length of \(Q''R''\) in \(\triangle Q''R''S''\): \(Q''(4,4)\) and \(R''(8,4)\), so \(Q''R''=\vert8 - 4\vert = 4\).
The scale factor \(k=\frac{Q''R''}{QR}=\frac{4}{2}=2\)

Answer:

\(\triangle QRS\) is translated \(8\) units to the right, then dilated by a scale factor of \(2\) with a center of dilation at the origin.