Question

latrell will rotate △abc 180° about one of the labeled points. latrell says, \the image will only touch the pre - image at exactly one point.\ which points could be his center of rotation? select each correct answer. point b point c point d point e

Explanation:

Step1: Understand the property of 180 - degree rotation

When a figure is rotated 180° about a point \(P\), a point \(Q(x,y)\) in the pre - image has an image \(Q'(-x + 2a,-y + 2b)\) if \(P=(a,b)\). If the center of rotation is a vertex of the triangle, the vertex will map to itself.

Step2: Analyze rotation about point \(B\)

If we rotate \(\triangle ABC\) 180° about point \(B\), point \(B\) maps to itself. Let's consider the other points. The image of \(\triangle ABC\) will have \(B\) as a common point. The rest of the points of the image and pre - image will not overlap.

Step3: Analyze rotation about point \(C\)

If we rotate \(\triangle ABC\) 180° about point \(C\), point \(C\) maps to itself. The image of \(\triangle ABC\) will have \(C\) as a common point. The rest of the points of the image and pre - image will not overlap.

Step4: Analyze rotation about point \(D\)

If we rotate \(\triangle ABC\) 180° about point \(D\), since \(D\) is on side \(AB\) (not a vertex). The image of the side \(AB\) (where \(D\) lies) after 180° rotation will overlap with the pre - image side \(AB\) along a line segment (not just a point).

Step5: Analyze rotation about point \(E\)

If we rotate \(\triangle ABC\) 180° about an interior point \(E\), the image and pre - image will overlap along a line segment (because the rotation of a non - vertex interior point will cause more than one point of overlap due to the nature of the triangle's interior).

Answer:

point \(B\), point \(C\)