Question

The points A(-2,2), B(4,4) and C(5,2) are the vertices of a triangle. The perpendicular bisector of AB and the line through A parallel to BC intersect at the point D. Find the area of the quadrilateral ABCD.?

i just draw its diagram so that its easy to find D coordinates

Answer 1

I get `77 .`

Explanation:

A(-2,2), B(4,4) and C(5,2)

I'm going to try without a figure.

Step 1. The perpendicular bisector of AB.

The midpoint of AB is `((-2+4)/2, (2+4)/2)=(1,3)`

The general line between `(a,b)` and `(c,d)` is `(c-a)(y-b)=(x-a)(d-b)`. The line between A and B is thus

`6(y-2) = 2(x + 2)`

`-2 x + 6y = text{constant}`

We don't care about the constant here because we're after the perpendicular through the midpoint, the perpendicular bisector. We swap the coefficients on `x` and `y`, negating one, and plug in the point for the constant:

`6x + 2y = 6(1) + 2(3) = 12`

Step 2. The line through A parallel to BC

BC has the equation

`(5 -4)(y-4) = (x-4)(2-4)`

`2x + y = text{constant}`

Again we don't care about the constant. We're after the parallel through A, so we plug in A to get the constant for the parallel:

`2x + y = 2(-2)+2 = -2`

Step 3. Find the meet D of the perpendicular bisector of AB and the line through A parallel to BC.

`6x + 2y = 12`

`2x + y=-2`

`4x + 2y = -4`

Subtracting,

`2x = 16`

`x = 8`

`y = -2 -2(8)= -18`

D(8,-18)

Step 4: Area of polygon A(-2,2), B(4,4), C(5,2), D(8,-18)

The Shoelace Theorem gives the area. It says the area is half the absolute value of the sum of cross products of successive sides. It's important to keep the order constant. The cross product of points is defined: `(a,b) times (c,d) = ad - bc.` When there are many points it helps to line up the points over their successor:

(-2,2), (4,4), (5,2), (8,-18)
(4,4), (5,2), (8,-18),(-2,2)

We just write down the area:

`A = 1/2 | -2(4) - 2(4) + 4(2)-4(5) + 5(-18) - 2(8) + 8(2) - (-18)(-2) |`

`A = 77`

This needs to be checked badly, but I gotta go.