Chapter 4.7, Problem 31HP

(a)

To determine

To explain:

Two transformations of Triangle 1 that can result in Triangle 2

Answer to Problem 31HP

Since each vertex and its image are the same difference from the y -axis, a reflection can map Triangle $2$ toTriangle $1$ .

Since each vertex and its image are in the same position, just 5 units to the right, a translation can mapTriangle $2$ to Triangle $1$ .

Explanation of Solution

Given Information:

  

Since each vertex and its image are the same difference from the y -axis, a reflection can map Triangle $2$ toTriangle $1$ .

Since each vertex and its image are in the same position, just 5 units to the right, a translation can mapTriangle $2$ to Triangle $1$ .

(b)

To determine

To explain:

If it must be true of the triangles in order for more than one transformation on a preimage to result in the sameimage.

Answer to Problem 31HP

The triangles must be either isosceles or equilateral. When triangles are isosceles or equilateral,they have a line of symmetry, so reflections result in the same figure.

Explanation of Solution

Given Information:

  

The triangles must be either isosceles or equilateral. When a triangle is isosceles or equilateral, areflection and a translation result in the same image.

They each have a line of symmetry so a reflection results in thesame figure as a translation.

By counterexample, consider this scalene triangle that has been translated.

  

Compare the image to the image from a reflection. The two images are not the same.