In square ABCD, E is the midpoint of BC, and F is the midpoint of CD. Let G be the intersection of AE and BF. Let M be the midpoint of AB, and let N be the intersection of AE and DM.

a) Find the ratios BG:MN, FG:MN, and DN:MN.

b) Compute the ratio of the area of GFDN to the area of GBMN.

Guest Sep 1, 2019

#1**+2 **

Here I will give you hint: EDIT: If you want you can go down to CPhill's answer for the answer

You can calculate the area of MAD, BEA, CFB.

You can calculate the area of BMDF by taking the area of the entire square and subtracting MAD and CFB from it.

You can do some coordinate geometry to find info about BGNM

CalculatorUser Sep 1, 2019

#2**+2 **

I like doing this in the following manner....construct a square with a side = 4 [ any side length would work ]

Let A = (0,0) B = (0,4) C = (4,4) and D = (4,0) E = (2,4) F = (4,2) M = (0, 2)

Here's a pic :

Triangles BCF and DAM are congruent right triangles

And the slope of BF = [ 4-2] / [ 0 - 4 ]= 2/-4 = -1/2

And the slope of EA = [ 4-0] / [2-0] = 4/2 = 2

So....these segements have reciprocal slopes so they are perpendicular

This means that triangles BCF and BGE are similar by AA congruency

BC = 4 CF = 2 BE = 2 BF = √ [ BF^2 + CF^2] = √ [4^2 + 2^2] = √20 = 2√5

So.....by similar triangles...... BG / BE = BC/ BF → BG/ 2 = 4/ [ 2√5 ] → BG = 4/√5

And GE = √[BE^2 - BG^2] = √[2^2 - (16/5] = √ 4 - (16/5] = √ [ 20 - 16] / √5 = 2/√5

And the area of triangle BCF = (1/2) BC * CF = 4

And since BG / BC = ( 4 / [ √5] ) / 4 = 1/ √5.....the area of triangle BGE = 4 (1/√5)^2 = 4/5

Likewise triangles MAD and MNA are similar and we can show that triangles BGE and ANM are congruent

So....BG/MN = BG / GE = [4/√5 ] / [2 /√5] = 2

And FG = BF - BG = 2√5 - 4/√5 = [10 √5 - 4√5] / 5 = 6√5/5 = 6/√5

So....FG/ MN = 6/√5 / [ 2/√5] = 6/2 = 3

And DN = DM - MN = BF - GE = 2√5 - 2/√5 = [ 10√5 - 2√5] / 5 = 8√5/5 = 8/√5

So DN / MN = [8/√5]/ [2/√5] = 8 /2 = 4

Area of BMDF = Area of the square less areas of BCF and DAM = 4^2 - 2 [4] = 16 - 8 = 8

Area of GBMN = area of BEA - areas of BGE and ANM = (1/2)BE * BA - 2(4/5) =

(1/2)(2)(4) - 8/5 =

4- 8/5 =

[20 - 8] / 5 =

12/5

So the area of GFDN = area of BMDF - area of GBMN = 8 - 12/5 = [40 - 12]/5 = 28/5

So GFDN / GBMN = [ 28/5] / [ 12/5] = 28/12 = 7 / 3

CPhill Sep 1, 2019