ABCD is a rectangle of dimensions 2 and 4 as shown in the figure above.
P and Q are points on the sides AB and CD, respectively, such that the straight line PQ is perpendicular to the diagonal DB and bisects it at point O.
Find the area of the quadrilateral CBOQ.
+128460 The area of the rectangle = 4 * 2 = 8 (1)
And the area of triangle ABD is 1/2 of this = 4 (2)
O is the center of the rectangle = (2,1)
The slope of the line containing DB = 2/4 = 1/2
So....the slope of the line containing QP = -2
And this line passes through O, so the equation of this line is
y = -2 ( x -2) + 1
y = -2x + 4 + 1
y= -2x + 5
We can find the x coordinate of Q by letting y = 0
0 = -2x + 5
-5 = -2x
x = -5/-2 = 2.5
So the area of triangle DOQ = (1/2) DQ * height = (1/2) (2.5) (1) = 1.25 (3)
So....the area of the yellow region = (1) -(2) - (3) =
8 - 4 - 1.25 =
8 - 5.25 =
2.75
