Question

in the pre - image of the parallelogram, the slope of ad is - 3 and the slope of bc is - 3. the image is translated down 7 units and then reflected over the y - axis.
the slope of ad is.
the slope of bc is.
what can you determine about the line segments ad and bc?
they are parallel.

Explanation:

Step1: Analyze the effect of translation and reflection on slope

Translation (moving down 7 units) does not change the slope of a line segment. Reflection over the \(y -\)axis changes the \(x -\)coordinate of points. The formula for slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). If we have a point \((x,y)\) and its reflection over the \(y -\)axis \((-x,y)\), for two - point \((x_1,y_1)\) and \((x_2,y_2)\) with slope \(m_1=\frac{y_2 - y_1}{x_2 - x_1}\), their reflections \((-x_1,y_1)\) and \((-x_2,y_2)\) have slope \(m_2=\frac{y_2 - y_1}{-x_2-(-x_1)}=-\frac{y_2 - y_1}{x_2 - x_1}=-m_1\).

Step2: Find the slope of \(B''C''\)

The slope of \(BC\) is \(- 3\). After translation (which doesn't affect slope) and reflection over the \(y -\)axis, if the original slope of a line segment is \(m\), the new slope \(m'=-m\). For \(BC\) with \(m = - 3\), after reflection over the \(y -\)axis (translation doesn't change slope), the slope of \(B''C''\) is \(m_{B''C''}=3\).

Step3: Find the slope of \(A''D''\)

The slope of \(AD\) is \(-3\). After translation (no - slope - change) and reflection over the \(y -\)axis, using the slope - reflection formula \(m'=-m\). For \(AD\) with \(m=-3\), the slope of \(A''D''\) is \(m_{A''D''}=3\).

Answer:

The slope of \(B''C''\) is \(3\). The slope of \(A''D''\) is \(3\).