Question

ABCD is a square. A line through B intersects CD
produced at E, the side AD at F and the diagonal AC at
GIF BG= 3 and GF = 1. Then find the length of FE.​

Answer 1

Answer:

8 units

Step-by-step explanation:

In ΔAGF and ΔCGB,

∠AGF = ∠CGB (vertically opposite angles)

AF || BC, so, ∠AFG = CBG (alternate interior angles)

So, ΔAGF ~ ΔCGB  (AA similarity)

So, GF/BG = GA/CG = AF/BC --------- (i)

Given, BG = 3, GF = 1.

So, GA/CG = 1/3

Let GA = x and CG = 3x

So, GA + CG = AC = 4x

Let the side of square be y.

So, diagonal = √(y^2 + y^2) = √2y (using pythagoras theorem with two sides)

So, AC = √2y = 4x  =>  y = 2√2 x.

BC = 2√2x,

From (i), AF/2√2x = 1/3  =>  AF = 2√2x/3

FD = AD - AF =  2√2x - 2√2x/3 = 4√2x/3

Now, in ΔFDE and ΔFAB,

∠FDE = ∠FAB = 90°

∠EFD = ∠BFA (vertically opposite angles)

So, ΔFDE ~ ΔFAB (AA similarity)

So, FD/FE = AF/FB

(4√2x/3)/FE = (2√2x/3)/4 = √2x/6

FE = 8 units.

Answer 2

Answer:9

Step-by-step explanation: