Question

a triangle on a coordinate plane has one vertex at (-7, -14). after a dilation with the origin as the center of dilation, the corresponding vertex of the dilated triangle is located at (-1, -2). which rule represents the dilation applied to the triangle? (a) (x, y)→(1/7 x, 1/7 y) (b) (x, y)→(1/2 x, 1/2 y) (c) (x, y)→(7x, 7y) (d) (x, y)→(2x, 2y)

Explanation:

Step1: Calculate the scale factor for \(x\) - coordinate

The original \(x\) - coordinate is \(x=-7\), and the new \(x\) - coordinate is \(x'=-1\). Let the scale factor be \(k\). Then \(x' = kx\), so \(k=\frac{x'}{x}=\frac{-1}{-7}=\frac{1}{7}\)

Step2: Calculate the scale factor for \(y\) - coordinate

The original \(y\) - coordinate is \(y = - 14\), and the new \(y\) - coordinate is \(y'=-2\). Using \(y'=ky\), we get \(k=\frac{y'}{y}=\frac{-2}{-14}=\frac{1}{7}\)

Answer:

A. \((x,y)\to(\frac{1}{7}x,\frac{1}{7}y)\)