Question

ABCD is a rectangle. the points E, F, G and H are chosen on the sides AB, BC, CD and DA respectively so EFGH is a rectangle. Furthermore CF = AH = 9m, DG = 15m. If AC is parallel to GH, show that ABCD is a square and find its side length.

there is the same question out there but explains that the side length of efgh is 16m, how does one acquire that answer?

Answer 1

Answer:

24 m

Step-by-step explanation:

CF = AH = 9 cm

So BF = DH = x xm

As AD = BC in rectangle

DG = 15cm

Let say CG = y cm

FG^2 = CF^2 + CG^2

FG^2 = 9^2 + y^2

FG = EH opposite sides of rectangle

EH^2 = AH^2 + AE ^2

9^2 + y^2 = 9^2 + AE^2

AE = y

Hence BE = 15cm as AB = CD

In triangle EBF & FCG

AngleB = angle C = 90deg

Angle BEF = X then angle BFE = 90-x

Angle CFG = x. And angle CGF = 90-x

Hence both triangles are similar

BF/ CG = BE/CF = EF/FG

x/y = 15/9

x = 5y/3

AC is parallel to GH

Triangle DGH & DAC are similar

x/(9+x). = 15/(15+y)

15x +xy = 135 + 15x

xy = 135

Putting value of x

(5y/3) y = 135

y^2 = 81

y = 9

x = 15

AD = 9+x = 9+15 = 24

DC = 15+y = 15+9 = 24

Hence ABCD is a square with side 24m