The slope-intercept equation for $L$ 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 $D$ .

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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.

Steps

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.

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