Question

triangle abc is reflected across the y - axis and then dilated by a factor of $\frac{1}{2}$ centered at the origin. which statement correctly describes the resulting image, triangle def?

a. neither the reflection nor the dilation preserves the side lengths and angles of triangle abc.
b. both the reflection and dilation preserve the side lengths and angles of triangle abc.
c. the dilation preserves the side lengths and angles of triangle abc. the reflection does not preserve side lengths and angles.
d. the reflection preserves the side lengths and angles of triangle abc. the dilation preserves angles but not side lengths.

Explanation:

Step1: Properties of reflection

A reflection is a rigid transformation. Rigid transformations preserve side - lengths and angles. So, when triangle \(ABC\) is reflected across the \(y\) - axis, the side - lengths and angles of the triangle are preserved.

Step2: Properties of dilation

A dilation with a scale factor \(k=\frac{1}{2}\) centered at the origin is a non - rigid transformation. For a dilation \((x,y)\to(kx,ky)\) (where \(k = \frac{1}{2}\) in this case). The formula for the length of a side of a triangle with vertices \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). After dilation, if we have two points \((x_1,y_1)\) and \((x_2,y_2)\) transformed to \((\frac{1}{2}x_1,\frac{1}{2}y_1)\) and \((\frac{1}{2}x_2,\frac{1}{2}y_2)\), the new distance \(d'=\sqrt{(\frac{1}{2}x_2-\frac{1}{2}x_1)^2+(\frac{1}{2}y_2 - \frac{1}{2}y_1)^2}=\frac{1}{2}\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). So, side - lengths are not preserved. But for angles, since the ratio of the sides of the dilated triangle to the original triangle is constant (by the property of similar triangles, \(\triangle ABC\sim\triangle DEF\) with a scale factor of \(\frac{1}{2}\)), angles are preserved (because similar triangles have congruent corresponding angles).

Answer:

D. The reflection preserves the side lengths and angles of triangle \(ABC\). The dilation preserves angles but not side lengths.