Question

1. in this diagram, point d is rotated ( 180^{circ} ) using b as the center. explain why the image of d must lie on the ray ba.

Explanation:

Step1: Recall the property of 180 - degree rotation

A \(180^{\circ}\) rotation about a center \(B\) maps a point \(D\) to a point \(D'\) such that \(B\) is the mid - point of the segment \(DD'\). That is, if we consider the line segment \(BD\) and its image \(BD'\) after a \(180^{\circ}\) rotation about \(B\), the points \(D\), \(B\), and \(D'\) are collinear.

Step2: Analyze the angles

Since \(\angle ABD+\angle CBD = 180^{\circ}\) (linear pair of angles, as \(A\), \(B\), \(C\) are collinear). When we rotate point \(D\) \(180^{\circ}\) about \(B\), the direction of the line segment \(BD\) is reversed. The ray \(BD\) is rotated to a ray such that the angle between the original ray \(BD\) and the image ray is \(180^{\circ}\).

Answer:

When a point \(D\) is rotated \(180^{\circ}\) about a center \(B\), the points \(D\), \(B\), and its image \(D'\) are collinear. The ray \(BD\) and the ray \(BA\) form a straight - line (since \(\angle ABD+\angle CBD=180^{\circ}\) and rotation of \(BD\) by \(180^{\circ}\) about \(B\) reverses its direction). So the image of \(D\) must lie on the ray \(BA\).