Find the area of a triangle given that midpoi | vedclass

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Question

(C) Let the vertices of the triangle be $A(x_1, y_1)$,$B(x_2, y_2)$,and $C(x_3, y_3)$.
The midpoints of the sides are given as $M_1(2, 7)$,$M_2(1, 1)

Vedclass · Accepted Answer

$94$

Vedclass · Answer

(C) Let the vertices of the triangle be $A(x_1, y_1)$,$B(x_2, y_2)$,and $C(x_3, y_3)$.
The midpoints of the sides are given as $M_1(2, 7)$,$M_2(1, 1)$,and $M_3(10, 8)$.
The area of the triangle formed by the midpoints of the sides of a triangle is $\frac{1}{4}$ of the area of the original triangle.
First,calculate the area of the triangle formed by the midpoints using the formula: $	ext{Area} = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$.
$	ext{Area}_{	ext{mid}} = \frac{1}{2} |2(1 - 8) + 1(8 - 7) + 10(7 - 1)|$.
$	ext{Area}_{	ext{mid}} = \frac{1}{2} |2(-7) + 1(1) + 10(6)|$.
$	ext{Area}_{	ext{mid}} = \frac{1}{2} |-14 + 1 + 60| = \frac{1}{2} |47| = 23.5$.
Since $	ext{Area}_{	ext{original}} = 4 	imes 	ext{Area}_{	ext{mid}}$,we have $	ext{Area}_{	ext{original}} = 4 	imes 23.5 = 94$.

Find the area of a triangle given that midpoints of its sides are $(2, 7)$,$(1, 1)$,and $(10, 8)$.

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