Question

which transformations map \\( \triangle jkl \\) onto \\( \triangle jkl \\)? select the two correct answers.
a a translation down 2 units, and then a rotation of \\( 180^\circ \\) about the origin
b a reflection over the x - axis, and then a translation up 2 units
c a rotation of \\( 90^\circ \\) counterclockwise about the origin, and then a translation up 2 units
d a translation down 2 units, and then a rotation of \\( 90^\circ \\) counterclockwise about the origin
e a translation down 2 units, and then a reflection over the x - axis
f a rotation of \\( 180^\circ \\) 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 F: A \(180^\circ\) rotation about the origin changes \((x,y)\) to \((-x,-y)\). After this rotation, a translation up 2 units (adding 2 to the \(y\)-coordinate) aligns the triangles.
• Option E: Translating down 2 units (subtracting 2 from \(y\)-coordinates) and reflecting over the \(x\)-axis (changing \((x,y)\) to \((x,-y)\)) also maps \(\triangle JKL\) to \(\triangle J'K'L'\).

Other options fail:

• A: Rotation after translation down 2 does not align.
• B: Reflection over \(x\)-axis + translation up 2 misaligns.
• C: \(90^\circ\) counterclockwise rotation + translation up 2 is incorrect.
• D: Translation down 2 + \(90^\circ\) counterclockwise rotation is incorrect.

Answer:

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