Problem
The slope-intercept equation for \( \mathrm{L} \) is \[ y=-2 x+b \] Since \( M(1,3) \) lies on \( L \), we have \( 3=-2(1)+b \). Hence, \( b=5 \), and the slope-intercept equation of \( L \) is \[ y=-2 x+5 \] Example 11> Prove analytically that the figure obtained by joining the midpoints of consecutive sides of a quadrilateral is a parallelogram. SOLUTION Locate a quadrilateral with consecutive vertices, \( A, B, C \), and \( D \) on a coordinate system so that \( A \) is the origin, \( B \) lies on the positive \( x \) axis, and \( C \) and \( D \) lie above the \( x \) axis. Let \( b \) be the \( x \) coordinate of \( B,(u, v) \) the coordinates of \( C \), and \( (x, y) \) the coordinates of \( D \).
Solution
Step 1 ：Let the midpoints of consecutive sides be E, F, G, and H. Then E is the midpoint of AB, F is the midpoint of BC, G is the midpoint of CD, and H is the midpoint of DA.
Step 2 ：Using the midpoint formula, we find the coordinates of E (\( \frac{b}{2}, 0 \)), F (\( \frac{b+u}{2}, \frac{v}{2} \)), G (\( \frac{u+x}{2}, \frac{v+y}{2} \)), and H (\( \frac{x}{2}, \frac{y}{2} \)).
Step 3 ：To prove that EFGH is a parallelogram, we need to show that opposite sides are parallel; EF is parallel to GH and EG is parallel to FH. We can use the slope formula to show this: slope of EF = slope of GH and slope of EG = slope of FH. Since the slopes are equal, we can conclude that EFGH is a parallelogram.
