Question

which of the following transformations maps xyz onto xyz? translation down 13 units translation down 12 units reflection across the x - axis reflection across the y - axis rotation 90° clockwise around the origin rotation 180° counter - clockwise around the origin rotation 180° around the origin

Explanation:

Step1: Analyze translation

Translation moves a figure without changing its orientation. If we consider a point \((x,y)\) in \(\triangle XYZ\) and translate it down \(n\) units, the new - point is \((x,y - n)\).

Step2: Analyze reflection across x - axis

For a point \((x,y)\) reflected across the \(x\) - axis, the new point is \((x,-y)\), which changes the orientation of the triangle.

Step3: Analyze reflection across y - axis

For a point \((x,y)\) reflected across the \(y\) - axis, the new point is \((-x,y)\), which changes the orientation of the triangle.

Step4: Analyze rotation

A \(90^{\circ}\) clock - wise rotation about the origin of a point \((x,y)\) gives \((y,-x)\), and a \(90^{\circ}\) counter - clockwise rotation about the origin of a point \((x,y)\) gives \((-y,x)\), both of which change the orientation of the triangle.

Step5: Check translation distances

If we assume a point in the upper - part of \(\triangle XYZ\) with a \(y\) - coordinate and a corresponding point in the lower - part of \(\triangle X'Y'Z'\) with a \(y\) - coordinate. By counting the grid squares between the corresponding points of the two triangles, we find that the translation is down 13 units.

Answer:

translation down 13 units