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Study Stuff for MCA Examinations - NIMCET
Study Stuff for MCA Examinations - NIMCET
Previous Year Question (PYQs)
If $(4,-3)$ and $(-9,7)$ are two vertices of a triangle and $(1,4)$ is its centroid, find the area of the triangle.
Solution
Let the third vertex be $(x,y)$. Centroid formula: $\displaystyle \left(\frac{4 + (-9) + x}{3}, \frac{-3 + 7 + y}{3}\right) = (1,4)$From x–coordinate:
$\dfrac{4 - 9 + x}{3} = 1$
$x - 5 = 3$
$x = 8$
From y–coordinate:
$\dfrac{-3 + 7 + y}{3} = 4$
$y + 4 = 12$
$y = 8$
So third vertex is:
$(8,8)$
Vertices:
$A(4,-3), B(-9,7), C(8,8)$
Area:
$\displaystyle \text{Area}
= \frac{1}{2}\left| x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2) \right|$
Substitute:
$\displaystyle
= \frac{1}{2}\left|4(7-8) + (-9)(8+3) + 8(-3-7) \right|$
$\displaystyle
= \frac{1}{2}\left| 4(-1) - 9(11) + 8(-10) \right|$
$\displaystyle
= \frac{1}{2}\left| -4 - 99 - 80 \right|$
$\displaystyle
= \frac{1}{2} \times 183$
$\displaystyle
= \frac{183}{2}$
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