Question

a composition of transformations maps \\( \triangle klm \\) to \\( \triangle klm \\).
the first transformation for this composition is \\_\\_\\_\\_\\_\\_, and the second transformation is a translation down and to the right.
\\( \circ \\) a \\( 90^\circ \\) rotation about point \\( l \\)
\\( \circ \\) a \\( 270^\circ \\) rotation about point \\( l \\)
\\( \circ \\) a \\( 90^\circ \\) rotation about point \\( p \\)
\\( \circ \\) a \\( 270^\circ \\) rotation about point \\( p \\)

Explanation:

Rotation is a transformation where a figure is turned around a fixed point. A \(90^{\circ}\) rotation clockwise (which is equivalent to a \(270^{\circ}\) rotation counter - clockwise) about a point changes the orientation of the figure.
Looking at the orientation of \(\triangle KLM\) and \(\triangle K'L'M'\), if we consider the general rule of rotation:

• For a \(90^{\circ}\) clockwise rotation (\(270^{\circ}\) counter - clockwise) about a point \(P\), the coordinates of a point \((x,y)\) (assuming the center of rotation is \((a,b)\)) follow the transformation rule \((x - a,y - b)\to(y - b,b-(x - a))\) (in a more intuitive sense, if we just look at the visual orientation change).

If we rotate \(\triangle KLM\) \(270^{\circ}\) about point \(P\), the orientation of the triangle changes in a way that matches the orientation of \(\triangle K'L'M'\) before the translation. A \(90^{\circ}\) rotation about \(L\) or \(P\) would not give the correct pre - translation orientation, and a \(270^{\circ}\) rotation about \(L\) is incorrect because the center of rotation for the first transformation (to get from \(\triangle KLM\) to the intermediate non - translated image) is \(P\) (by visual inspection of the figure's orientation change).

Answer:

a \(270^{\circ}\) rotation about point \(P\)