Question 1042781: Given the rectangle ABCD, with tan angle x= 3/4 and tan angle y= 1/6, the area of the shaded region is what fraction of the area of the rectangle?
http://i.imgur.com/VnVdLtb.png
Found 2 solutions by LinnW, solver91311:
Answer by LinnW(1048)   (Show Source):
You can put this solution on YOUR website! assume that the side CD = 3 and side AC = 4.
This is consistent with Tan x = 3/4.
Since AC = 4, BC is also = 4
With Tan y = 1/6, this means length of BE = 2/3.
Tan y = 1/6, with adjacent side = 4, opposite side must be 2/3.
The area of the shaded area is
the area of triangle ABC - the area of triangle EBC, or
(1/2)(4*3) - (1/2)(4*(2/3))
6 - (1/2)(8/3)
6 - 8/6
36/6 - 8/6
28/6
14/3
4 2/3
The area of the whole rectangle is 4*3 = 12,
so the ratio of the shaded to the whole is (4 2/3)/12
(14/3)/12
7/18 , ratio of shaded to whole
Answer by solver91311(24713)  (Show Source):
You can put this solution on YOUR website!
Since the actual size of the rectangle is arbitrary, we can use the fact that the tangent of x is 3/4 to assign the value 4 to the measure of the side of the rectangle that is adjacent to angle x, and the value 3 to the measure of the side opposite angle x. The other long side of the rectangle must also be 4, and since 4 is 2/3 of 6, assign 2/3 (2/3 times 1) to the segment that is opposite angle y.
Then, relatively speaking, the area of the entire rectangle is 12 square units, the area of the triangle containing angle x is 6 square units, and the area of the triangle containing angle y is 4/3 square units. The sum of the areas of the two triangles is then 22/3 and 22/3 divided by 12 is 11/18.
Then the shaded area is 1 minus 11/18. 7/18
John
My calculator said it, I believe it, that settles it
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