The coordinates of the endpoints of AB and CD are (Ax1, Ay1), (Bx2, By2), (Cx3, Cy3), and (Dx4, Dy4). Which condition proves that AB || CD? A. (Ax1 – Bx2) / (Ay1 – By2) = (Cx3 – Dx4) / (Cy3 – Dy4) B. (Ax1 – Cx3) / (Ay1 – Cy3) = (Bx2 – Dx4) / (By2 – Dy4) C. (Ax1 – Dx4) / (Ay1 – Dy4) = (Bx2 – Cx3) / (By2 – Cy3) D. (Ay1 – By2) / (Ax1 – Bx2) = (Cy3 – Dy4) / (Cx3 – Dx4)
The Correct Answer and Explanation is:
The correct answer is: D. (Ay1 – By2) / (Ax1 – Bx2) = (Cy3 – Dy4) / (Cx3 – Dx4)
Explanation:
To determine whether two line segments, AB and CD, are parallel, we compare their slopes. Two lines are parallel if and only if they have the same slope (assuming they are not vertical lines).
The slope of a line segment connecting two points (x1,y1)(x_1, y_1)(x1,y1) and (x2,y2)(x_2, y_2)(x2,y2) is calculated by:slope=y2−y1x2−x1\text{slope} = \frac{y_2 – y_1}{x_2 – x_1}slope=x2−x1y2−y1
Now, let’s apply this to the segments:
- Segment AB has endpoints (Ax1,Ay1)(Ax1, Ay1)(Ax1,Ay1) and (Bx2,By2)(Bx2, By2)(Bx2,By2).
Its slope is: mAB=By2−Ay1Bx2−Ax1m_{AB} = \frac{By2 – Ay1}{Bx2 – Ax1}mAB=Bx2−Ax1By2−Ay1 - Segment CD has endpoints (Cx3,Cy3)(Cx3, Cy3)(Cx3,Cy3) and (Dx4,Dy4)(Dx4, Dy4)(Dx4,Dy4).
Its slope is: mCD=Dy4−Cy3Dx4−Cx3m_{CD} = \frac{Dy4 – Cy3}{Dx4 – Cx3}mCD=Dx4−Cx3Dy4−Cy3
For AB and CD to be parallel:mAB=mCD⇒By2−Ay1Bx2−Ax1=Dy4−Cy3Dx4−Cx3m_{AB} = m_{CD} \Rightarrow \frac{By2 – Ay1}{Bx2 – Ax1} = \frac{Dy4 – Cy3}{Dx4 – Cx3}mAB=mCD⇒Bx2−Ax1By2−Ay1=Dx4−Cx3Dy4−Cy3
However, option D is just a rearrangement using equivalent expressions (the slopes written from the first point to the second, and signs flipped):Ay1−By2Ax1−Bx2=Cy3−Dy4Cx3−Dx4\frac{Ay1 – By2}{Ax1 – Bx2} = \frac{Cy3 – Dy4}{Cx3 – Dx4}Ax1−Bx2Ay1−By2=Cx3−Dx4Cy3−Dy4
Since:Ay1−By2Ax1−Bx2=−(By2−Ay1)−(Bx2−Ax1)=By2−Ay1Bx2−Ax1\frac{Ay1 – By2}{Ax1 – Bx2} = \frac{-(By2 – Ay1)}{-(Bx2 – Ax1)} = \frac{By2 – Ay1}{Bx2 – Ax1}Ax1−Bx2Ay1−By2=−(Bx2−Ax1)−(By2−Ay1)=Bx2−Ax1By2−Ay1
the slopes remain equal. Therefore, option D correctly expresses the condition for AB and CD to be parallel.
Options A, B, and C do not compare the slopes of AB and CD directly, and thus do not prove parallelism between the two segments.
