Line segments are drawn from the vertices of the large square to the midpoints of the opposite sides to form a smaller, white square.
If each red or blue line-segment measures 10 m long, what is the area of the smaller, white square in m^2?
+128474 Note that at the top of the figure we have a right triangle with two legs of 10 and 20
So the area of this right triangle = (1/2) (10)(20) = 100
And the base of this triangle = sqrt [ 10^2 + 20^2] = sqrt [ 500] = 10sqrt(5)
So......we can find the height of this triange as
100 = (1/2) (10sqrt (5) ) height
20 / sqrt (5) = height = 4sqrt (5)
And using similar triangles the height of the small right triangle on the top left = 1/2 of this = 2sqrt (5) = sqrt (20)
And the base of this triangle = sqrt [ 10^2 - (sqrt(20))^2 ] = sqrt [ 100 - 20] = sqrt (80)
So the area of this small right triangle = (1/2)sqrt (20)sqrt(80) = (1/2)sqrt(1600) = (1/2) 40 = 20
And we have two of these triangles in the larger right triangle at the top
So....the area of the remaining trapezoid = 100 - 2(20) = 60
So....we have 4 smaller right triangles and 4 trapezoids....so their combined areas =
4 ( 20 + 60) = 4 (80) = 320 (1)
So....the area of the white square =
area of the square - (1) =
20^2 - 320 =
400 - 320 =
80 (m^2)
