A Triangle in the xy-Coordinate Plane Has Vertices With Coordinates (7, 0), (0, 8), and (20, 10) GMAT Problem Solving

Question: A triangle in the xy-coordinate plane has vertices with coordinates (7, 0), (0, 8), and (20, 10). What is the area of this triangle?

1. 72
2. 80
3. 87
4. 96
5. 100

‘A triangle in the xy-coordinate plane has vertices with coordinates (7, 0), (0, 8), and (20, 10). What is the area of this triangle?’ - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide”. 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:
It is given in the question that a triangle in the xy-coordinate plane has vertices with coordinates (7, 0), (0, 8), and (20, 10).
It has asked to find out the area of the triangle.
This question is from the coordinate geometry.
Firstly plot the triangle in the cartesian plane.
Firstly we’ll consider the area of the trapezoid formed by the sides,
(0,0), (8,0) , (20,0), (20,10).
The area of the trapezoid = ((8+13) / 2)*20 = 180
Now let the triangle formed by the sides (7,0), (20,0), and (20,10) be T1.
Area of the triangle T1 = ½ *(13*10) = 65
Now let the triangle formed by the sides (7,0), (0,0), and (8,0) be T2.
Area of T2 = ½*8*7 = 28
The required area will be equal to the area of the trapezoid - the sum of areas of two triangles.
Required area = 180 -65-28 =87
Therefore the correct answer will be option C.

Correct Answer: C

Approach Solution 2:

The points are plotted in the diagram and a rectangle is formed which is marked in red color.
The area of the triangle will be equal to the area of the square - (the sum of areas of marked 3 triangles)
area of rectangle: 20x10 = 200
area of 1: 8*7/2=28
area of 2: 13*10/2=65
area of 3: 2*20/2=20
Area of required triangle = 200 - 28-20 -65 = 87
Therefore the correct answer will be option C.

Correct Answer: C

Approach Solution 3:

The area of triagle formula is A = bh / 2. If the triangle is circumscribed by a rectangle, as given in the below diagram, the areas of the three surrounding triangles can be subtracted from the rectangle’s area to get the result.

Case 1, the triangle to the lower left has b = 7 and h = 8, so its area is (7)(8) / 2 = 28.
Case 2, the triangle to the lower right has b = 13 and h = 10, so its area is (13)(10) / 2 = 65.
Case 3, the topmost triangle has b = 20 and h = 2, so its area is (20)(2) / 2 = 20.

The total area of the surrounding triangle is 28 + 65 + 20 = 113 square units.
The total area of the rectangle is 20 x 10 = 200 square units

Hence, the triangle’s area is 200 – 113 = 87 square units.

Correct Answer: C

Suggested GMAT Problem Solving Questions:

• What is the remainder when 333^222 is divided by 7? GMAT Problem Solving
• In a college of 300 students, every student reads 5 newspapers GMAT Problem Solving
• What is the Largest Power of 3 Contained in 200 GMAT Problem Solving
• Find The Value Of x GMAT Problem Solving
• 4 Out Of 15 Apples Are Rotten GMAT Problem Solving
• Last Year 26 Members of a Certain Club Traveled to England GMAT Problem Solving
• Running at the Same Constant Rate, 6 Identical Machines GMAT Problem Solving
• A conference room is equipped with a total of 45 metal or wooden chairs GMAT Problem Solving
• A welder received an order to make a 1 million litre cube-shaped tank GMAT Problem Solving
• After 6 games, Team B had an average of 61.5 points per game GMAT Problem Solving
• If 12 ounces of a strong vinegar solution are diluted with 50 ounces of water GMAT Problem Solving
• A contractor estimated that his 10-man crew could complete the construction GMAT Problem Solving
• A circle is inscribed in a square with the diagonal of 4 centimeters GMAT Problem Solving
• At a dog competition, a dog is awarded 10 points if it runs through 4 pipes GMAT Problem Solving
• Two consultants, Mary and Jim, can type up a report in 12.5 hours and edit it in 7.5 hours GMAT Problem Solving
• Machine A can do a certain job in 12 days working 2 full shifts GMAT Problem Solving
• The square of \(5^\sqrt2)\ =? GMAT Problem Solving
• If 4 people are selected from a group of 6 married couples GMAT Problem Solving
• If the equation |x|+|y|= 5 encloses a certain region on the graph GMAT Problem Solving
• If x = ¾ and y = ⅖ , what is the value of (x^2+ 6x+ 9) - (y^2-2y+ 1)? GMAT Problem Solving