Question: ABCD is a trapezium, in which AD and BC are parallel. If the four sides AB, BC, CD, DA are 9,12, 15, 20. Find the magnitude of the sum of the squares of the two diagonals.

1. 638
2. 786
3. 838
4. 648
5. 726

Correct Answer: B
Solution and Explanation:
Approach Solution 1:

The problem statement states that:

Given:

• ABCD is a trapezium, where AD and BC are parallel.
• AB=9, BC=12, CD=15, DA=20.

Find out:

• The sum of the squares of the two diagonals.

Let the height of the trapezium i.e the distance between the parallel sides be h.
Let the distance between C` and D be x.
Let the distance between B` and A be y

C` and B` are projections of B and C on AD.

Therefore, if this trapezium is constructed to scale, \(\angle A \)is an obtuse angle, that's why what projects onto what is slightly different, but it doesn't impact the calculations that follow.
Indeed, 20 = 12 + x + y,

Therefore, x + y = 8 —-- (i)

By applying Pythagorean Theorem on the side triangles, we get:

x^2 + h^2 = 9^2 = 81 —--(ii)
y^2 + h^2 = 15^2 = 225 —--(iii)

By applying Pythagorean Theorem for the diagonals, we get:

(AC)^2 = h^2 + (20 -x)^2
= h^2 + 400 - 40x + x^2
= (h^2 + x^2) + 400 - 40x

From equation (ii), we get:

(AC)^2 = 81 + 400 - 40x
(AC)^2 = 481 - 40x

Moreover, we get:

(BD)^2 = h^2 + (20 -y)^2
= h^2 + 400 - 40y + x^2
= (h^2 + y^2) + 400 - 40y

From equation (iii), we get:

(BD)^2 = 225 + 400 - 40y
(BD)^2 = 625 - 40y

Therefore, the sum of squares of diagonals = (AC)^2 + (BD)^2
= 481 - 40x + 625 - 40y
= 1106 - 40(x + y)
= 1106 - 40(8) —- [from equation (i), we get x + y = 8]
= 1106 - 320 = 786

Approach Solution 2:

The problem statement suggests that:

Given:

• ABCD is a trapezium, where AD and BC are parallel.
• AB=9, BC=12, CD=15, DA=20.

Find out:

• The sum of the squares of the two diagonals.

As per the formula, we know that:

Sum of the squares of the diagonals = Sum of the squares of the non-parallel sides + 2*product of parallel sides

Therefore, in this case, sum of the squares of the two diagonals = 9^2 + 15^2 + 2* 20* 12 = 786.

“ABCD is a trapezium, in which AD and BC are parallel. If the four sides”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This topic has been taken from the book “McGraw-Hill's Conquering the GMAT Math”. The GMAT Problem Solving questions test the candidates’ skill in mathematical calculations in solving numerical problems. GMAT Quant practice papers enable the candidates to get familiar with several sorts of questions that will improve their mathematical learning.

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