What is the Area of the Triangle with the following Vertices L(1,3) M(5,1) and N(3,5)?

Question: What is the area of the triangle with the following vertices L(1,3) M(5,1) and N(3,5)?

1. 3
2. 4
3. 5
4. 6
5. 7

“What is the area of the triangle with the following vertices L(1,3) M(5,1) and N(3,5)?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT quant topics in the problem-solving part require calculative mathematical problems that should be solved with proper mathematical knowledge.

Answer:

Approach Solution 1-

Vertices of the triangle are: L(1,3) M(5,1) and N(3,5).

Using these vertices, plot a graph as shown in the figure below:

We can see that a red triangle is formed by joining all those points. Now make a square around this triangle.

Firstly, find the area of the square formed in the figure as:

Length of the side of square = 4

Area of square = \((side)^2\)= \(4^2\)= 16

Now, we will find the area of the corner right-angled triangles formed outside the red triangle and inside the square:

Area of a triangle =\(\frac{1}{2}*B*H\)

Hence, from the figure drawn above: -

Area of Triangle (1) = \(\frac{1}{2}*B*H\)= \(\frac{1}{2}*2*2\)= 2

Area of Triangle (2) = \(\frac{1}{2}*B*H\) = \(\frac{1}{2}*2*4\) = 4

Area of Triangle (3) = \(\frac{1}{2}*B*H\)= \(\frac{1}{2}*2*4\) = 4

Hence, Total Area of Triangle = 2 + 4 + 4 = 10

Now, we will calculate the Area of Red Triangle

So, Area of Red Triangle = Area of square – Area of corner triangles

Area of Red Triangle = 16 – 10 = 6

Final Answer = 6

Correct option: D

Approach Solution 2-

In this method, we will directly use the formula to calculate the area of the triangle.

Write down the original vertices of the triangle:

\((x_1,y_1)\)= (1,3)

\((x_2,y_2)\)= (5,1)

\((x_3,y_3)\)= (3,5)

Now plot the graph using these vertices:

If the vertices of a triangle are given

Then Area of \(\bigtriangleup{LMN}\)= \(\frac{1}{2}|x_1(y_2,y_3)+x_2(y_3,y_1)+x_3(y_1,y_2)\)

Area of \(\bigtriangleup{LMN}\) = \(\frac{1}{2}| 1 (1 – 5) + 5 (5 – 3) + 3 (3 – 1)|\)

Area of \(\bigtriangleup{LMN}\)= \(\frac{1}{2}|1 (-4) + 5 (2) + 3 (2)|\)

Area of  \(\bigtriangleup{LMN}\)= \(\frac{1}{2}|- 4 + 10 + 6\)|

Area of \(\bigtriangleup{LMN}\)= \(\frac{1}{2}|12|\)

Area of  \(\bigtriangleup{LMN}\)= 6

Correct option: D

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