In rectangle ABCD, I is the midpoint of AD and E and F trisect BC. BD intersects EI and FI and G and H. Find the ratio of the area of triangle GHI to the area of rectangle ABCD.

Guest Dec 13, 2020

#1**+2 **

In rectangle ABCD, I is the midpoint of AD and E and F trisect BC. BD intersects EI and FI and G and H. Find the ratio of the area of triangle GHI to the area of rectangle ABCD.

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Since a square is also a rectangle, we can use it to make life easier.

Let the square side be 3 units.

ID = 1.5 EF = 1 ∠EIF = 2* tan^{-1}[(EF/2) / AB] = 18.92464442º

∠ADB = 45º ∠EID = 90 + (∠EIF / 2) = 99.4623222º ∠FID = 90 - (∠EIF / 2) = 80.5376778º

Now you have enough information to calculate the area of triangles GID and HID.

But if you're lazy like me, you can visit this website: https://www.triangle-calculator.com/

jugoslav Dec 13, 2020

#2**+3 **

Here's a solution where the rectangle really is a rectangle and not necessarily a square.

Run horizontal lines GG' and HH' from G and H across to AB.

G' and H' will be used to write down the heights of the triangles GID and HID.

Triangles BEG and DIG are similar so \(\displaystyle \frac{EG}{GI}=\frac{BE}{ID}=\frac{1/3}{1/2}=\frac{2}{3}.\)

So BG' = (2/5)AB, and G'A = (3/5)AB,

so the area of the triangle DIG = (3/5)AB times (1/2)AD times (1/2) = (3/20) area of the rectangle.

Triangles BFH and DIH are similar so \(\displaystyle \frac{FH}{HI}=\frac{BF}{ID}=\frac{2/3}{1/2}=\frac{4}{3}.\)

So BH' = (4/7)AB and H'A = (3/7)AB,

so the area of the triangle DIH = (3/7)AB times (1/2)AD times (1/2) = (3/28) area of the rectangle.

Area of the triangle GHI is then (3/20) - (3/28) = (3/70) area of the rectangle.

Tiggsy Dec 13, 2020