Y 10

1

B(7,9)

Create a rectangle around the parallelogram. The

dimensions of this rectangle are

Find the area of the four right triangles surrounding the

parallelogram. The total area of the triangles is

square units

Subtract the triangle areas from the area of the

rectangle to obtain the area of the parallelogram. The

area of parallelogram ABCD is square units.

6

2

-10

6

-6

A-3,-1)

-2

10

C(8, -1)

D(-2,-11)

Step-by-step explanation:

See below for a graph showing the parallelogram and the rectangle.

The dimensions of the rectangle are 11 by 20. The rectangle area is ...

  rectangle area = bh = (11)(20) = 220 . . . square units

There are 2 right triangles with a base of 10 and a height of 1. There are 2 right triangles with a base of 10 and a height of 10. The total area of the triangles is ...

  triangle area = 2(1/2)(10)(1) +2(1/2)(10)(10) = 10(1 +10) = 110 . . . square units

The area of the parallelogram is the difference of these, ...

  parallelogram area = rectangle area - triangle area

  = (220 units²) -(110 units²) = 110 units²

The area of parallelogram ABCD is 110 square units.

_____

Check

The parallelogram has a base of 10√2 and a height of 5.5√2, so an area of (10√2)(5.5√2) = 110 square units. This matches the above result.

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4