Question

part d mrs. smith gives a group of students a list of five different transformations of triangle 3 to draw as follows: transform triangle 3 into triangle 4 by rotating 90° counterclockwise around the point (1,1). transform triangle 3 into triangle 5 using the rule (x,y)→(2x, -y). transform triangle 3 into triangle 6 using the rule (x,y)→(-y, -x - 2). transform triangle 3 into triangle 7 by translating 3 units right, then reflecting across the line y = x. transform triangle 3 into triangle 8 using the rule (x,y)→(1/3x, 1/3y - 3). which of triangles 2 through 8 are congruent to triangle 1? provide evidence, including why each triangle is or is not congruent to triangle 1, to support your answers.

Explanation:

Step1: Recall congruence and transformation rules

Congruent triangles have same shape and size. Transformations like translation, rotation, reflection preserve congruence, while dilation (scaling) may not.

Step2: Analyze each transformation

Transformation to Triangle 4

A 90 - degree counter - clockwise rotation around a point is a rigid transformation, so Triangle 3 and 4 are congruent.

Transformation to Triangle 5

The rule $(x,y)\to(2x, - y)$ is a non - rigid transformation (a dilation in the x - direction and a reflection in the x - axis). Since there is a dilation factor of 2 in the x - direction, Triangle 3 and 5 are not congruent.

Transformation to Triangle 6

The rule $(x,y)\to(-y,-x - 2)$ represents a rotation and a translation. Rigid transformations, so Triangle 3 and 6 are congruent.

Transformation to Triangle 7

Translation and reflection are rigid transformations. So Triangle 3 and 7 are congruent.

Transformation to Triangle 8

The rule $(x,y)\to(\frac{1}{3}x,\frac{1}{3}y - 3)$ is a dilation (scaling by a factor of $\frac{1}{3}$ in the x and y directions with a vertical shift) which changes the size, so Triangle 3 and 8 are not congruent.
Triangles 5 and 8 are not congruent to Triangle 1 (since they are not congruent to Triangle 3 and if Triangle 3 is congruent to Triangle 1, non - congruent to 3 means non - congruent to 1).

Answer:

Triangles 5 and 8 are not congruent to Triangle 1. Evidence: Transformation to Triangle 5 involves a non - rigid transformation (dilation in x - direction) and transformation to Triangle 8 involves a dilation (scaling in x and y directions), which change the size of the triangle, and non - congruent to Triangle 3 implies non - congruent to Triangle 1 if Triangle 1 and 3 are congruent.