GMAT Club World Cup 2022 (DAY 4): A trapezoid shown above, is divided

Answer is A

AB is longer than CD, so that CD/AB must be smaller than 1. Only A satisfies.

IF AB>CD THEN ALWAYS CD/AB IS 0<CD/AB>1
SO ONLY OPTION A IS SATISFY
ANSWER IS A

Given that CD>AB
Means CD/AB <1 (Since AB & CD are distances hence positive Numbers)
Only Option A <1, remaining options are greater than 1
Logical Option selection, no calculations needed, process of Elimination

Correct answer : Choice A

Given AB > CD.

To Find CD/AB

intutively speaking: if the denominator is bigger than the numerator, then the given equation CD/AB is a proper fraction.
For a proper fraction, the value should always be less than 1

choice B is greater than 1 - Wrong
choice C is greater than 1 - Wrong
choice D is greater than 1 - Wrong
choice E is greater than 1 - Wrong
In the options given , only choice A = 2 - root(2) = 2- 1.4 = .6 is less than 1. hence correct answer !.

Since ab is greater than cd, cd/ab must be less than 1. Only option (a) satisfies this.
Hence the answer

I might be bizarrely wrong about this, but it's said that AB > CD. And we need to find CD/AB.

If the numerator is lesser than the denominator, the answer is less than one. And since, they both are sides of the triangle, they have to be positive.

Only option A is less than one. And I think that's the answer.

Given: A trapezoid is divided into 4 parts having the areas of 2, 3, 4, and 5 cm^2, as shown above.
Asked: If AB > CD, what is the value of CD/AB ?

Let the value of CD/AB be x
CD/AB = x

Altitude from point E on AB & CD meet on points F & G respectively
Let EF = h1 & EG=h2

Area of trapezoid ABCD = (AB+CD)*(h1+h2)/2 = 2+3+4+5 = 14 cm^2
(AB + CD) (h1 + h2) = 28 cm^2 (1)

Area of triangle ABE = AB*h1/2 = 4
AB*h1 = 8 cm^2 (2)

Area of triangle CDE = CD*h2/2 = 2
CD*h2 = 4 cm^2 (3)

Eq (3) / Eq (2)
(CD/AB) * (h2/h1) = 4/8 = 1/2
h1/h2 = 2x ; h1 = 2xh2 ; h2 = h1/2x (4)

(AB + CD) (h1 + h2) = ABh1 + CDh2 + ABh2 + CDh1 = 28
8 + 4 + ABh1/2x + CD2xh2 = 28
8/2x + 8x = 16
1/x + 2x = 4
2x^2 - 4x + 1 = 0
\(x =\frac{ (4 + \sqrt{4^2 - 4*2})}{2*2} = \frac{(4 + 2\sqrt{2})}{4} = \frac{(2 + \sqrt{2})}{2} = 1 + \frac{1}{\sqrt{2}} = 1.707\)
Since x > 1

IMO D

Bunuel None of the answers seem correct; Please check and correct. Option D seems closest and should be divided by 2.

Answer A


Since AB>CD, can I shamelessly say the ratio CD:AB will be less than 1 and all options other than A are greater than 1?
Will this be accepted as a valid explanation Bunuel ?

Easy trick:
Since it is given that CD<AB, CD/AB<1. All options other than Option A are greater than 1 and are hence illogical.

The answer is D (2+ sqrt.2).

The area of the trapezium is 1/2 * (sum of the parallel side) * the distance between them= 1/2 *(AD+CD)* (h1+h2) = 14, where h1 is the height of traingle with are 4 and h2 with area 2. h1 = h2 = height of trapezium.

Also, 1/2*AB*h1 = 4 & 1/2 * CD*h2 = 2.

Solving these, we will get CD/AB = 2+sqrt 2. Opiton D

If \(CD < AB\), then \(\frac{CD}{AB}\) should be less than 1. Only option A is less than 1.

Hence, the answer is A.

Since we are asked for a ratio and we know CD<AB thus the ratio has to be less than 1 . Only B gives that

Ans B

OA = A
Let's say the centre point in trapezoid is O

so, area of triangle AOB = 4
now area of triangle AOB = (base * latitude)/2
base = AB , latitude = x

so AB*x = 8 , x = 8/AB

now, area of triangle COD = 2
area of triangle COD = (base*latitude)/2 = (CD*y)/2
so CD*y = 4 , y = 4/CD

now area trapezoid ABCD = total area of four triangles = 2 + 3 + 4 + 5 = 14

area trapezoid ABCD = height of trapezoid * (AB+CD)/2 = (x+y)*(AB+CD)/2

so xAB +xCD + yAB + yCD = 28
8 + xCD + yAB + 4 = 28
xCD + yAB = 16
CD*(8/AB) + AB*(4/CD) = 16
8*(CD/AB) + 4(AB/CD) = 16

let say AB/CD = k
so, 8/k + 4k = 16
2/k + k = 4
\( k^2 -4k + 2 = 0\)

now roots for this equation will be k=AB/CD = \(2+\sqrt{2} \)or \(2-\sqrt{2}\)

but AB>CD is given so root must be greater than 1 , so CD/AB must be less than 1
so ans will be CD/AB= \(2-\sqrt{2}\)

AB>CD, so ratio must be less than 1. only A is possible answer
All other choices are more than 1

Given : AB>CD. That means CD/AB is less than 1. Only option A is less than 1.

We know that AB > CD

Question: what is the value of CD/AB

We know AB is +ve (as it represents length) , hence we can divide both sides of the equation by AB

\(\frac{CD}{AB}\) < 1

Let's look at the options -

A. 2−√2 -- Less than 1

B. √2 -- 1.414

C. 2 -- 2

D. 2+√2 -- greater than 2

E. 2+2√2 --greater than 2

IMO A

It is given that AB>CD. So, CD/AB<1. Only answer choice A is less than 1. Hence, answer A.