The sides of a triangle are 9, 12, 15. The area of triangle formed by its incentre, centroid and orthocentre is:

\frac{5}{2}

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\frac{7}{2}

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\frac{3}{2}

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\frac{1}{2}

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Solution

The correct option is C $\frac{3}{2}$
Let $a = 9, b = 12, c = 15$
Clearly $a^{2} + b^{2} = c^{2} \Rightarrow$ Triangle is right angled
Let $A = (0, 12), B = (9, 0), C = (0, 0)$
Thus centroid, $G = (\frac{0 + 9 + 0}{3}, \frac{12 + 0 + 0}{3}) = (3, 4)$
Orthocentre $O = (0, 0)$ , since triangle is right angled at $C$
Incentre $I = (\frac{a x_{1} + b x_{2} + c x_{3}}{a + b + c}, \frac{a y_{1} + b y_{2} + c y_{3}}{a + b + c}) = (3, 3)$
Hence area of $△ G O I = \frac{1}{2} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} 0 & 0 & 1 3 & 4 & 1 3 & 3 & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ = \frac{3}{2}$ .

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Similar questions

(- 3, 5, 2), (6, 2, 5)

I. The centroid of the triangle formed by $(2, 3, - 1), (5, 6, 3), (2, - 3, 1)$ is	a) $(2, 2, 2)$
II. The circumcentre of the triangle formed by $(1, 2, 3), (2, 3, 1), (3, 1, 2)$ is	b) $(3, 1, 4)$
III. The orthocenter of the triangle formed by $(2, 1, 5), (3, 2, 3), (4, 0, 4)$ is	c) $(1, 1, 0)$
IV. The incentre of the triangle formed by $(0, 0, 0), (3, 0, 0), (0, 4, 0)$ is	d) $(3, 2, 1)$
	e) $(0, 0, 0)$

(1, 1)

(3, 2)

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