Question

the graph below shows two triangles. triangle abc is the image of triangle abc formed by dilation. which statement is not true about the relationship between the two triangles? a. the side lengths of abc and abc are proportional. b. triangle abc is similar to triangle abc by a scale factor of 2. c. the angle measures of abc and abc are congruent. d. triangle abc is similar to triangle abc by a scale factor of $\frac{1}{2}$.

Explanation:

Step1: Recall dilation properties

Dilation creates similar - figures. For similar figures, side - lengths are proportional and angle - measures are congruent. If $\triangle A'B'C'$ is the image of $\triangle ABC$ formed by dilation, and we observe the size of the triangles, $\triangle A'B'C'$ is smaller than $\triangle ABC$.

Step2: Determine the scale factor

The scale factor $k$ of a dilation from $\triangle ABC$ to $\triangle A'B'C'$ is the ratio of the side - lengths of $\triangle A'B'C'$ to $\triangle ABC$. Since $\triangle A'B'C'$ is smaller, the scale factor $k=\frac{1}{2}$, not $2$.

Step3: Analyze each option

• Option A: For similar triangles (formed by dilation), side - lengths are proportional. This is true.
• Option B: $\triangle A'B'C'$ is smaller than $\triangle ABC$, so the scale factor is $\frac{1}{2}$, not $2$. This statement is false.
• Option C: In a dilation, angle - measures of the pre - image and the image are congruent. This is true.
• Option D: Since $\triangle A'B'C'$ is smaller than $\triangle ABC$, $\triangle A'B'C'$ is similar to $\triangle ABC$ by a scale factor of $\frac{1}{2}$. This is true.

Answer:

B. Triangle $A'B'C'$ is similar to triangle $ABC$ by a scale factor of $2$.