Question

1. triangle xyz is dilated using the origin as the center of dilation to create triangle xyz. what rule best represents the dilation applied to triangle xyz to create triangle xyz? a (x, y)→(3x, 6y) b (x, y)→(6x, 3y) c (x, y)→(3x, 3y) d (x, y)→(6x, 6y)

Explanation:

Step1: Recall dilation rule

A dilation centered at the origin with scale - factor \(k\) has the rule \((x,y)\to(kx,ky)\). In a dilation, all coordinates of the original figure are multiplied by the same non - zero scale factor \(k\).

Step2: Analyze options

Option A \((x,y)\to(3x,6y)\) has different scale factors for \(x\) and \(y\) (\(3\) for \(x\) and \(6\) for \(y\)), so it is not a uniform dilation. Option B \((x,y)\to(6x,3y)\) also has different scale factors for \(x\) and \(y\). Option C \((x,y)\to(3x,3y)\) has a uniform scale factor of \(3\) for both \(x\) and \(y\). Option D \((x,y)\to(6x,6y)\) has a scale factor of \(6\) for both \(x\) and \(y\). Since in a dilation centered at the origin, the transformation should multiply both the \(x\) and \(y\) coordinates by the same number, we need to check the figure (visually, if the sides of the dilated triangle are all \(3\) times the length of the original triangle's sides, for example). Assuming a uniform dilation from the figure, if we assume the scale factor is \(3\), the rule \((x,y)\to(3x,3y)\) is correct.

Answer:

C. \((x,y)\to(3x,3y)\)