Question

Line Segment BC has endpoints B (3,5) and C (7,15). Find the missing coordinates of A (x,9) and D (17,y) such that AB and CD are perpendicular to BC.

Answer 1

Answer:

x=-7

y=11

Step-by-step explanation:

Perpendicular Vectors

Two vectors defined as their endpoints $\vec u=$ and $\vec v=$ are perpendicular if their dot product a.b is zero. The dot product is

$\vec u.\vec v=ac+bd$

In other words

ac+bd=0

Let's treat all the points as the extremes of vectors, so we can easily find the missing coordinates

B (3,5) and C (7,15) define a segment, the vector  

$\overrightarrow{BC}==$

The point A is A (x,9), we need to form a vector with B

$\overrightarrow{AB}==$

this vector must be perpendicular to BC, so, applying the dot product we have

$4(x-3)+40=0$

$4x-12=-40$

$x=-7$

The point D is D(17,y), we need to form a vector with C

$\overrightarrow{CD}==$

this vector must be perpendicular to BC, so, applying the dot product we have

$(4)(10)+(10)(y-15)=0$

$40+10y-150=0$

$10y=110$

$y=11$

The points are

$\boxed{A(-7,9), D(17,11)}$