Question

which transformations map △uke onto △jkl? select the two correct answers.
a a translation down 2 units, and then a rotation of 180° about the origin
b a reflection over the x - axis, and then a translation up 2 units
c a rotation of 90° counterclockwise about the origin, and then a translation up 2 units
d a translation down 2 units, and then a rotation of 90° counterclockwise about the origin
e a translation down 2 units, and then a reflection over the x - axis
f a rotation of 180° about the origin, and then a translation up 2 units

Explanation:

To determine the transformations mapping \(\triangle JKL\) to \(\triangle J'K'L'\), we analyze each option:

• Option A: Translate down 2, then rotate \(180^\circ\) about origin. A \(180^\circ\) rotation changes \((x,y)\) to \((-x,-y)\), but the orientation and position don’t match after these steps.
• Option B: Reflect over \(x\)-axis (changes \((x,y)\) to \((x,-y)\)) then translate up 2. This aligns the triangle’s orientation and position with \(\triangle J'K'L'\).
• Option C: Rotate \(90^\circ\) counterclockwise (changes \((x,y)\) to \((-y,x)\)) then translate up 2. The \(90^\circ\) rotation distorts the shape’s orientation, so this is incorrect.
• Option D: Translate down 2, then rotate \(90^\circ\) counterclockwise. The \(90^\circ\) rotation misaligns the triangle, so incorrect.
• Option E: Translate down 2, then reflect over \(x\)-axis. Reflecting after translating down 2 would not match the target triangle’s position.
• Option F: Rotate \(180^\circ\) about origin (changes \((x,y)\) to \((-x,-y)\)) then translate up 2. A \(180^\circ\) rotation followed by upward translation aligns the triangle with \(\triangle J'K'L'\).

Thus, the correct transformations are B (reflect over \(x\)-axis, then translate up 2) and F (rotate \(180^\circ\) about origin, then translate up 2).

Answer:

B. a reflection over the x - axis, and then a translation up 2 units
F. a rotation of \(180^\circ\) about the origin, and then a translation up 2 units