GMAT Problem Solving - The Sides of a Quadrilateral Taken in Order are 16,16,14 and 10

Topic: The sides of a quadrilateral taken in order are 16,16,14 and 10. The angle contained by the first two sides is 60 degrees. What is the area of the quadrilateral?

1. 12\(\sqrt{252}\)
2. 104\(\sqrt{3}\)
3. 16\(\sqrt{3}\)(4+\(\sqrt{15}\))
4. 14\(\sqrt{3}\)+5
5. 40\(\sqrt{3}\)

“The Sides of a Quadrilateral Taken in Order are 16,16,14 and 10” - is a topic of the GMAT Problem Solving of GMAT. This question has been taken from the book "GMAT Official Guide Quantitative Review". To solve GMAT Problem Solving questions a student must have knowledge about a good number of qualitative skills. The GMAT Quant section consists of 31 questions in total. The GMAT quant topics in the problem-solving part require calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation:

Approach Solution 1:

This problem has one approach to answer

Explanation: Given to us that the sides of the quadrilateral taken in order are 16,16,14 and 10. The angle contained by the first two sides is 60 degrees. We are required to find out the area of the quadrilateral.
To solve this question observe that the first two sides are of length 16 and 16 and they make 60 degrees with each other.
This gives the sign of an equilateral triangle. An equilateral triangle is a triangle having all the sides equal and each angle of 60 degrees.
Therefore when we’ll draw a diagonal the triangle that will form will be an equilateral triangle.
First we'll find the area of the equilateral triangle.
It should be known by the candidate that the equilateral triangle having side a has an area of A = \(\sqrt{3}\)/4 * a2
For this triangle, the side of the triangle = 16
So A = \(\sqrt{3}\)/4 * 162
A = \(\sqrt{3}\)*16 * 4
A = 64\(\sqrt{3}\) sq units
For the second triangle let base b = 16
The height of the second triangle will separate this triangle into two parts which are both right-angled. Let x be the shorter side and (16 - x) be the longer side of the base.
Now we’ll use the Pythagoras theorem,
It states that for a right angled triangle,
l^2 = b^2 + h^2
Where l = length of the hypotenuse
b = base of the triangle
H =height of the triangle.
Putting the values we get
h^2 = 10^2 - x^2

Also h^2 = 14^2 - (16 - x)^2

100 - x^2 = 14^2 - (16 - x)^2
100 - x^2 = 196 - (256 - 32 x + x^2)
100 - x^2 = 196 - 256 + 32x - x^2
100 = -60 + 32x
32x = 160

X = \(\frac{160}{32}\)

X = 5
Height h^2 = 10^2 - 5^2 = 100 -25 = 75
H = 5\(\sqrt{3}\)
Area of the second triangle =\(\frac{1}{2}\)* b * h = \(\frac{1}{2}\) * 5\(\sqrt{3}\) * 16 = 40\(\sqrt{3}\)
The area of the quadrilateral will be the sum of the areas of the triangles.
Area of the quadrilateral = 40\(\sqrt{3}\) + 64\(\sqrt{3}\) = 104\(\sqrt{3}\)
Therefore option B will be the correct answer.

Suggested GMAT Problem Solving Samples

1. Find The Value Of x GMAT Problem Solving 
2. 4 Out Of 15 Apples Are Rotten GMAT Problem Solving
3. Last Year 26 Members of a Certain Club Traveled to England GMAT Problem Solving
4. Running at the Same Constant Rate, 6 Identical Machines GMAT Problem Solving 
5. A conference room is equipped with a total of 45 metal or wooden chairs GMAT Problem Solving
6. A welder received an order to make a 1 million litre cube-shaped tank GMAT Problem Solving
7. After 6 games, Team B had an average of 61.5 points per game GMAT Problem Solving
8. If 12 ounces of a strong vinegar solution are diluted with 50 ounces of water GMAT Problem Solving
9. A contractor estimated that his 10-man crew could complete the construction GMAT Problem Solving
10. A circle is inscribed in a square with the diagonal of 4 centimeters GMAT Problem Solving
11. The square of 5√252 =? GMAT Problem Solving 
12. After 6 Games, Team B Had an Average of 61.5 Points Per Game. GMAT Problem Solving 
13. If 12 Ounces of a Strong Vinegar Solution are Diluted with 50 Ounces GMAT Problem Solving 
14. A Contractor Estimated that his 10-Man Crew GMAT Problem Solving 
15. A Circle is Inscribed in a Square with the Diagonal of 4 Centimeters.  GMAT Problem Solving 
16. At a Dog Competition, a Dog is Awarded 10 Points if it Runs GMAT Problem Solving 
17. The Population of the Bacteria Colony Doubles Every Day GMAT Problem Solving
18. If tu=xytu=xyand ty=uxty=ux Where t, u, x, and y are Non-Zero Integers GMAT Problem Solving 
19. A Farm has Chickens, Cows and Sheep GMAT Problem Solving 
20. Machine A can do a Certain Job in 12 Days Working 2 Full Shifts GMAT Problem Solving