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Question

Let A(0,−6), B(1,2), C(3,−3), D(−3,2), and E(−4,−2) be given points, then which of the following is (are) true?

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Solution

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A(0,-6), B(1,2), C(3,-3),D(-3,2) and E(-4,-2)

Area of the pentagon = Area formed by a polygon =

∣∣∣(x1y2−y1x2)+(x2y3−y2x3)...+(xny1−ynx1)2∣∣∣

where a1 is the coordinate of vertex 1 and yn is the y-coordinate of the nth vertex etc.

Area ∣∣
∣
∣
∣∣[(−3×2)−(2×1)]+[(1×−3)−(3×2)]+[(3×−6)−(−3×0)]+[(−4×−2)−(−2×−3)]+[(0×−2)−(−6×−4)]2∣∣
∣
∣
∣∣=∣∣∣−732∣∣∣=732

Area of triangle formed by points below $x-axis,

Area=12bh=×−6=242=12 square unit

Angle subtended by BD at A=tan−1(8261)

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