Question

two transformations are applied to the triangle abc in the grid. abc is the result of a horizontal stretch, while abc is the result of a reflection. which of the following statements is true?

a. the horizontal stretch preserves the distances between points in the original shape, but it does not preserve the angles formed by the sides. the reflection preserves distances as well as angles.
b. the horizontal stretch neither preserves the distances between points in the original shape nor preserves the angles formed by the sides. the reflection preserves angles but does not preserve distances.
c. the horizontal stretch does not preserve the distances between points in the original shape, but it preserves the angles formed by the sides. the reflection preserves distances but does not preserve angles.
d. the horizontal stretch neither preserves the distances between points in the original shape nor preserves the angles

Explanation:

• Horizontal stretch:
• A horizontal stretch changes the \(x -\)coordinates of the points. For example, if a point \((x,y)\) is stretched horizontally by a factor \(k\), it becomes \((kx,y)\).
• This changes the distances between points (since the \(x -\)coordinates are scaled). Also, the angles formed by the sides are changed. For instance, consider a right - angled triangle. A horizontal stretch will distort the right - angle.
• Reflection:
• A reflection is an isometry. An isometry is a transformation that preserves distances and angles. For any two points \(P(x_1,y_1)\) and \(Q(x_2,y_2)\) and their images \(P'(x_1',y_1')\) and \(Q'(x_2',y_2')\) under a reflection, the distance \(d(P,Q)=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) and \(d(P',Q')=\sqrt{(x_2' - x_1')^2+(y_2' - y_1')^2}\) are equal. Also, for any two line segments \(AB\) and \(BC\) in a figure and their images \(A'B'\) and \(B'C'\) under a reflection, the angle \(\angle ABC=\angle A'B'C'\).

Answer:

D. The horizontal stretch neither preserves the distances between points in the original shape nor preserves the angles formed by the sides. The reflection preserves distances as well as angles.