Question

1. in the diagram below, △abc is the image of △abc after a transformation.

describe the transformation that was performed.
explain why △abc ~ △abc.

1. the vertices of △jkl have coordinates j(5, 1), k(−2, −3), and l(−4, 1). under which transformation is the image △jkl not congruent to △jkl?

a. a translation of two units to the right and two units down
b. a counterclockwise rotation of 90 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 △abc is dilated by a scale factor of 3, which statement is true of the image △abc?

a. 3ab = ab
b. bc = 3bc
c. m∠a = 3m∠a
d. m∠c = m∠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:

Step1: Analyze Option A (Translation)

A translation (two units right, two units down) is a rigid transformation. Rigid transformations preserve side lengths and angles, so \(\triangle J'K'L'\) would be congruent to \(\triangle JKL\).

Step2: Analyze Option B (Rotation)

A counterclockwise rotation of \(90^\circ\) around the origin is a rigid transformation. Rotations preserve side lengths and angles, so \(\triangle J'K'L'\) would be congruent to \(\triangle JKL\).

Step3: Analyze Option C (Reflection)

A reflection over the \(x\)-axis is a rigid transformation. Reflections preserve side lengths and angles, so \(\triangle J'K'L'\) would be congruent to \(\triangle JKL\).

Step4: Analyze Option D (Dilation)

A dilation with a scale factor of \(2\) and center at the origin is a non - rigid transformation. Dilation changes the side lengths (scales them by the factor of \(2\)) while preserving angles. So, \(\triangle J'K'L'\) will be similar (but not congruent) to \(\triangle JKL\) because the side lengths are not preserved (they are doubled), violating the congruence condition (corresponding sides must be equal in length for congruence).

Answer:

D. a dilation with a scale factor of 2 and centered at the origin