Question

15 multiple choice 1 point which of the following will map △abc onto △abc? reflection in the x - axis translation 2 units right and 1 unit up reflection in the y - axis clockwise turn 90° about the origin

Explanation:

Step1: Analyze reflection in the x - axis

If we reflect a point \((x,y)\) in the \(x\) - axis, the transformation is \((x,y)\to(x, - y)\). This would change the \(y\) - coordinate signs. But looking at the figure, the orientation of the triangles (not just coordinate signs) does not match a reflection over the \(x\) - axis.

Step2: Analyze translation 2 units right and 1 unit up

If we consider a translation \((x,y)\to(x + 2,y+1)\). Let's assume a point \(A(x_1,y_1)\) in \(\triangle ABC\). The shape and relative positions of the triangles (for example, the side - length relationships and angles) do not match a simple translation. A translation just slides the figure without rotation or reflection.

Step3: Analyze reflection in the y - axis

If we reflect a point \((x,y)\) in the \(y\) - axis, the transformation is \((x,y)\to(-x,y)\). If we assume \(A\) has coordinates \((x_1,y_1)\), \(B\) has coordinates \((x_2,y_2)\) and \(C\) has coordinates \((x_3,y_3)\) in \(\triangle ABC\), the \(x\) - coordinates of the corresponding points \(A'\), \(B'\) and \(C'\) in \(\triangle A'B'C'\) have opposite signs (if we assume \(A\) is at \((- 1,1)\), \(A'\) is at \((1,1)\); if \(B\) is at \((-3,1)\), \(B'\) is at \((1,1)\) (wait, no - actually, using the property that for a reflection over the \(y\) - axis, the distance of a point from the \(y\) - axis is preserved but the \(x\) - coordinate sign changes. The triangles \(\triangle ABC\) and \(\triangle A'B'C'\) are mirror - images about the \(y\) - axis.

Step4: Analyze clockwise turn \(90^{\circ}\) about the origin

The transformation for a clockwise \(90^{\circ}\) rotation about the origin is \((x,y)\to(y,-x)\). If we check the side - length ratios and the orientation of the triangles (the slope of the sides), the triangles do not match the properties of a \(90^{\circ}\) clockwise rotation. For example, the length of \(AB\) and the length of \(A'B'\) (if we calculate using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)) and their orientation (the angle between the sides) do not match a \(90^{\circ}\) rotation.

Answer:

reflection in the \(y\) - axis