State the following statement is True or False
The area (in sq. units) of the triangle formed by the points with polar coordinates $(1, 0) (2, \frac{π}{3})$ and $(3, \frac{2 π}{3})$ , is $\frac{5 \sqrt{3}}{4}$

True

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False

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Solution

The correct option is A True
Given points $(1, 0)$ , $(2, \frac{π}{3})$ and $(3, \frac{2 π}{3})$
Area of triangle, $△ = \frac{1}{2} | r_{1} r_{2} sin (θ_{2} - θ_{1}) + r_{2} r_{3} sin (θ_{3} - θ_{2}) + r_{3} r_{1} sin (θ_{1} - θ_{3}) |$
$= \frac{1}{2} ∣ ∣ ∣ 2 sin (\frac{π}{3}) + 6 sin (\frac{π}{3}) + 3 sin (- \frac{2 π}{3}) ∣ ∣ ∣$
$= \frac{1}{2} ∣ ∣ ∣ ∣ 8 (\frac{\sqrt{3}}{2}) - 3 (\frac{\sqrt{3}}{2}) ∣ ∣ ∣ ∣$
$= \frac{5 \sqrt{3}}{4}$

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