In right triangle $ABC$, $AB=9$, $BC=13$, and $\angle B = 90^\circ$. Points $D$ and $E$ are midpoints of $\overline{AB}$ and $\overline{AC}$ respectively; $\overline{CD}$ and $\overline{BE}$ intersect at point $X$. Compute the ratio of the area of quadrilateral $AEXD$ to the area of triangle $BXC$.
Here's one way to do this
Here's a pic :
DE/AD = CB/AB
DE/4.5 =13/9
DE = 13(4.5)/9 = 6.5
So ....the area of triangle ADE = (1/2)(DE*AD = (1/2)6.5 * 4.5 = 14.625 units^2
And we can find X as the intersection of two lines
The first has the equation y = (4.5 /6.5)x ⇒ y = (9/13)x (1)
D= (0, 4.5) C = (13, 0) so the slope of the line connecting these two points is
-4.5 /13 = -9/26
So....the line through these these two points is y = (-9/26)x + 4.5 (2)
Setting (1) = (2) to find the x cooordinate of X we have
(9/13)x = (-9/26)x + 4.5
[(18/26) + (9/26) ] x = 4.5
(27/26)x = 9/2
x = (26/27)(9/2) = 13/3
And y = (9/13)(13/3) = 3
So.....the area of triangle DXE = (1/2) (4.5 - 3)(6.5) = (1/2) (1.5)(6.5) = 4.875 units^2
So the area of AEXD = area of triangle ADE + area of triangle DXE = 14.625 + 4.875 = 19.5 units^2
And the area of triangle BXC = (1/2)(13)(3) = 39/2 = 19.5 units^2
So the ratio of the areas = 19.5 / 19.5 = 1 ..... exactly the same areas!!!
Here's another way to do this with similar triangles....it is much easier
First....calculate DE as I did before = 6.5
Note that angle BXC = angle DXE (vertical angles)
Angle DEX = Angle CBX (alternate interior angles between parallels )
So triangle BXC is isimilar to triangle EXD
Since base ED is 1/2 of BC
Then the height of triangle EXD is 1/2 that of triangle BXC
So y = 4.5 is split into tree equal parts....and the height of BXC is (2/3)of these = 3
Then area of triangle EXD = 1/4 area of triangle BXC
Area of BXC = (1/2)(3)(13) = 39/2 = 19.5
And the area of EXD = (39/8)
And the area of triangle ADE is (1/2)(4.5)(6.5) = (117/8)
So the area of AEXD = area of triangle EXD + area of triangle of triangle ADE =
(39/8 + 117/8) = (156/8) = 19.5
So the ratios of BXC / AEXD = 19.5 /19.5 = 1