NIMCET 2012 — Mathematics PYQ

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Question

NIMCET 2012 — Mathematics PYQ

NIMCET | Mathematics | 2012

If $(4, -3)$ and $(-9, 7)$ are the two vertices of a triangle and $(1, 4)$ is its centroid, then area of triangle is:

Choose the correct answer:

Explanation

Solution

Find the third vertex $(x_3, y_3)$ :

Given vertices are $A(4, -3)$ and $B(-9, 7)$ . Centroid $G$ is $(1, 4)$ .

Using Centroid Formula: $G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right)$

For x-coordinate:

$1 = \frac{4 + (-9) + x_3}{3} \implies 3 = -5 + x_3 \implies x_3 = 8$

For y-coordinate:

$4 = \frac{-3 + 7 + y_3}{3} \implies 12 = 4 + y_3 \implies y_3 = 8$

Third vertex $C$ is $(8, 8)$ .
Calculate the Area of Triangle:

Using the formula: $\text{Area} = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$

Substitute $A(4, -3)$ , $B(-9, 7)$ , and $C(8, 8)$ :

$\text{Area} = \frac{1}{2} |4(7 - 8) + (-9)(8 - (-3)) + 8(-3 - 7)|$

$\text{Area} = \frac{1}{2} |4(-1) - 9(11) + 8(-10)|$

$\text{Area} = \frac{1}{2} |-4 - 99 - 80|$

$\text{Area} = \frac{1}{2} |-183|$

$\text{Area} = \frac{183}{2}$

Correct Option: (c)

Choose the correct answer:

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