If ABCD is a square with area 625, and CEFD is a rhombus wit

Area of shaded region = Area of square - (Area of rhombus - Area of triangle MFD)



Area of square = 625 so side = 25
Area of rhombus = 500. So altitude = 500/25 = 20
MF = root (25^2 - 20^2) = 15
Area of triangle MFD = (1/2) * 15 * 20 = 150

Area of shaded region = 625 - (500 - 150) = 275

Kudos to Karishma. Failed to apply some basic theorems.

Went to calculation of diagonals of rhombus

GMAT questions are lot easier than you think !!

Dear Karishma,

how do you get "MF = root (25^2 - 20^2) = 15"?

Area of rhombus is Base*Height = 500
Height = 500/25 = 20 = MD

Now MFD is right angled triangle so using pythagorean theorem,
MD^2 + MF^2 = FD^2
MF^2 = FD^2 - MD^2
MF = sqrt(25^2 - 20^2)
MF = sqrt(625 - 400)
MF = sqrt(225)
Mf = 15

I don't understand. Is there a different perspective with which we are looking at the diagram in this solution? The one in question is from a different angle, right?

Hi Karishma

Can you please tell How did we arrive at FD=25 ?

Thank you

hi

I am not Karishma mam the great, but I will try to help you

please remember that the sides of a rhombus are always equal
here you can see the side "CD", which is a side of both the square and the rhombus

so, as the side of the square is 25, the side of the rhombus is also 25

hope this helps!

thanks

Nice explanation.

VeritasPrepKarishma

The only thing that threw me off in this problem is the fact that when I read, on a different website, it said that the figure is not drawn to scale? I had initially made the assumption that the base of the rhombus was equal to the length of the square and derived the height using that- but then I second guessed myself and try to work backwards from the area of the rhombus in order to find the lengths of the diagonals and found myself in a mess. In this problem we can make the assumption that the base of the rhombus is equal to a side length of the square? That seems to be a valid takeaway from initially glancing at the problem

It is not an assumption from the figure. You are given that ABCD is a square so CD is a side of the square. You are also given that CEFD is a rhombus so CD is a side of the rhombus too. Hence the base of the rhombus is equal to the length of the side of the square.

The area of the rhombus is given by base*height
Since area =500=25*h therefore h=20
now we can find the hypotenuse of the triangle =15
area of the triangle is =150
So 500-150=350
The area of the shaded region =625-350=
275

Area of square = 625 so side = 25
Area of rhombus = 500. So altitude = 500/25 = 20
MF = root (25^2 - 20^2) = 15
Area of triangle MFD = (1/2) * 15 * 20 = 150

Area of shaded region = 625 - (500 - 150) = 275

in this why are we separately subtracting the area of the triangle.. ?? is it not already a part of the rhombus??

(1st) notice that the the bottom side CD is shared by both the rhombus and the square

(2nd) we can find the side of the “square” that would be hidden behind the rhombus by taking the square root of the area = 625

Side CD = sqrt(625) = 25


(2nd) we are given that the area of the rhombus = 500

Area of rhombus = (Base side) * (Perpendicular Height)

We can draw 2 perpendicular height lines:

(1) drawn from vertex E perpendicular to side CD ——> call the intersection point X

(2) drawn from vertex D to side EF of the rhombus ———> call the intersection point Y

The area of the shaded portion = area of square - (area of right triangle CED) - (area of rectangle EXDY)

(3rd) find the length of the perpendicular heights = EX and DY ——> call H

Area of rhombus = 500 = (side CD) * (H)

500 = 25H

H = 20 = EX = DY

(4th)we can find the length of CX and XD, which together sum to equal side CD of 25, using Pythagorean Theorem

Let the leg CX = l

Using right triangle CEX:

(25)^2 = (20)^2 + (l)^2

(l))^2 = 225

l = 15 ——> which means

CX = 15

And

DX = CD - CX

DX = 25 - 15

DX = 10


(4th) we can find the area of the right triangle CEX and the rectangle EXYD now:

Area right triangle CEX = (1/2) (15) (20) = 150

Area of rectangle EXYD = (10) (20) = 200

150 + 200 = 350

350 is the area covering what would be the square ABCD


Area of shaded region = 625 - 350 = 275


(E)275

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