In a Trapezium ABCD, AB Is Parallel To DC, AB = 3*DC, And The Diagonals Of The Trapezium Intersect At O GMAT Problem Solving

Question: In a trapezium ABCD, AB is parallel to DC, AB = 3*DC, and the diagonals of the trapezium intersect at O. The ratio of the area of ∆OCD to the area of ∆OAB is:

1. 1:9
2. 1:3
3. 3:1
4. 1:10
5. Cannot be determined

“In a trapezium ABCD, AB is parallel to DC, AB = 3*DC, and the diagonals of the trapezium intersect at O.” -is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Advanced Quant”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation:

Approach Solution 1:

There is only one approach to solve this problem.
Given:

• In a trapezium ABCD, AB is parallel to DC
• AB = 3*DC
• Diagonals of the trapezium intersect at O.

Find Out:

• The ratio of the area of ∆OCD to the area of ∆OAB is

In Triangle AOB and Triangle COD
Angle AOB = Angle COD (Since they are Vertical Angles)
Other two angles are formed between a pair of parallel line AB || DC
Angle BAO = Angle DCO (The Alternate interior angle)
Angle ABO = Angle CDO (The Alternate interior angle)
Therefore, by AAA Property the two triangle are similar
The ratio of a side is already given, i.e. 1:3
The ratio of their areas will be square of the ratio of their sides (Property of similar triangles)
Hence, 1^2 : 3^2 = 1:9

Correct Answer: A

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