Consider the points $A ( 0,4 ) , B ( 3,0 ) , C ( 4,7 ) \text {, and } D ( 7,3 )$ . Let $L _ { 1 }$ be the segment connecting the midpoint of segment AB to the midpoint of segment BC. Let $L _ { 2 }$ be the segment connecting the midpoint of segment BC to the midpoint of segment CD. Is $L _ { 1 }$ perpendicular to $L _ { 2 }$ ?
A) Yes
B) No

Question

Quizplus · Accepted Answer

We first find the coordinates of the midpoints:
Midpoint of AB: $M_1 = \left(\frac{0+3}{2},\frac{4+0}{2}ight) = (1.5,2)$
Midpoint of BC: $M_2 = \left(\frac{3+4}{2},\frac{0+7}{2}ight) = (3.5,3.5)$
Midpoint of CD: $M_3 = \left(\frac{4+7}{2},\frac{7+3}{2}ight) = (5.5,5)$
Now we find the slopes of the two lines:
$L_1:$ Slope of $AB$: $\frac{4-0}{0-3} = -\frac43$. Slope of $BC$: $\frac{7-0}{4-3} = 7$
Thus, slope of $L_1$ is $\frac{-1}{\frac73} = -\frac37$
$L_2:$ Slope of $BC$: $\frac{7-0}{4-3} = 7$. Slope of $CD$: $\frac{3-7}{7-4} = -\frac43$
Thus, slope of $L_2$ is $\frac{7}{-\frac43} = -\frac{21}{4}$
Since the product of the slopes is $-\frac37\cdot -\frac{21}{4}= \frac{21}{4} 
eq -1$, the two lines are not perpendicular.

Consider the Points A(0,4),B(3,0),C(4,7), and D(7,3)A ( 0,4 ) , B ( 3,0 ) , C ( 4,7 ) \text {, and } D ( 7,3 )A(0,4),B(3,0),C(4,7), and D(7,3)

Consider the Points $A ( 0,4 ) , B ( 3,0 ) , C ( 4,7 ) \text {, and } D ( 7,3 )$