Question

1. in the diagram below, $\triangle abc$ is the image of $\triangle abc$ after a transformation.

describe the transformation that was performed.
explain why $\triangle abc \sim \triangle abc$.

1. the vertices of $\triangle jkl$ have coordinates $j(5,1)$, $k(-2,-3)$, and $l(-4,1)$. under which transformation is the image $\triangle jkl$ not congruent to $\triangle jkl$

a. a translation of two units to the right and two units down
b. a counterclockwise rotation of 180 degrees around the origin
c. a reflection over the x - axis
d. a dilation with a scale factor of 2 and centered at the origin

1. if $\triangle abc$ is dilated by a scale factor of 3, which statement is true of the image $\triangle abc$

a. $3ab = ab$
b. $bc = 3bc$
c. $m\angle a = 3(m\angle a)$
d. $3(m\angle c) = m\angle c$

1. in the diagram below, $\overline{cd}$ is the image of $\overline{ab}$ after a dilation of scale factor $k$ with center $e$.

which ratio is equal to the scale factor $k$ of the dilation?
a. $\frac{ec}{ea}$
b. $\frac{ba}{ea}$
c. $\frac{ea}{ba}$
d. $\frac{ea}{ec}$

Explanation:

Question 19

Step1: Recall Dilation Properties

Dilation is a transformation that scales a figure by a scale factor. Corresponding sides are proportional (scale factor times original side), and corresponding angles are congruent (angle measure doesn't change).

Step2: Analyze Each Option

• Option A: \( 3A'B' = AB \) would imply \( A'B'=\frac{1}{3}AB \), but dilation with scale factor 3 means \( A'B' = 3AB \), so A is false.
• Option B: By dilation property, corresponding sides are scaled by the scale factor. So \( B'C' = 3BC \), which matches the scale factor 3.
• Option C: Angles are congruent in dilation, so \( m\angle A' = m\angle A \), not \( 3(m\angle A) \), so C is false.
• Option D: Angles are congruent, so \( m\angle C' = m\angle C \), not \( 3(m\angle C') = m\angle C \), so D is false.

Step1: Recall Congruence Transformations

Translations, rotations, and reflections are rigid transformations (preserve side lengths and angles), so they produce congruent figures. Dilation (unless scale factor 1) changes side lengths, so the image is similar but not congruent.

Step2: Analyze Each Option

• Option A: Translation is rigid, so congruent.
• Option B: Rotation (180°) is rigid, so congruent.
• Option C: Reflection over x - axis is rigid, so congruent.
• Option D: Dilation with scale factor 2 changes side lengths (scales them by 2), so \( \triangle J'K'L' \) is not congruent to \( \triangle JKL \).

Step1: Recall Dilation Scale Factor

In a dilation with center E, the scale factor \( k \) is the ratio of the length of the image segment to the length of the original segment, or the ratio of the distances from the center to the image point and the center to the original point. For points A (original) and C (image), and center E, \( k=\frac{EC}{EA} \) because \( EC \) is the distance from E to the image point C, and \( EA \) is the distance from E to the original point A.

Step2: Analyze Each Option

• Option A: \( \frac{EC}{EA} \) is the ratio of the distance from E to image (C) and E to original (A), which is the scale factor for dilation.
• Option B: \( \frac{BA}{EA} \) is not related to dilation from E.
• Option C: \( \frac{EA}{BA} \) is not related to dilation from E.
• Option D: \( \frac{EA}{EC} \) would be the reciprocal of the scale factor.

Answer:

B. \( B'C' = 3BC \)

Question 18