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Coordinate Geometry

Short Answer Type

171.

The coordinates of the vertices of ∆ABC are A(4, 1), B(-3, 2) and C(0, k). Given that the area of ABC is 12 unit², find the value of k.

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172. Find the area of the ∆ABC with vertices A(-5, 7), B(-4, -5), and C(4, 5).

Given points are A(-5, 7), B(-4, 5) and C(4, 5).
Here, we have
x₁ = -5,    y₁ = 7
x₂ = -4,    y₂ -5
and    x₃ = 4,    y₃ = 5
Area of ∆ABC

$equals 1 half left square bracket straight x subscript 1 left parenthesis straight y subscript 2 minus straight y subscript 3 right parenthesis plus straight x subscript 2 left parenthesis straight y subscript 3 minus straight y subscript 1 right parenthesis plus straight x subscript 3 left parenthesis straight y subscript 1 minus straight y subscript 2 right parenthesis right square bracket equals 1 half left square bracket negative 5 left parenthesis negative 5 minus 5 right parenthesis plus left parenthesis negative 4 right parenthesis left parenthesis 5 minus 7 right parenthesis plus 4 left parenthesis 7 plus 5 right parenthesis right square bracket equals 1 half left square bracket negative 5 straight x minus 10 plus left parenthesis negative 4 straight x minus 2 right parenthesis plus 4 space straight x space 12 equals 1 half left square bracket 50 space plus space 8 space plus space 48 right square bracket equals 1 half left square bracket 106 right square bracket equals 53 space sq. space units$
Hence area of the ∆ABC = 53 sq. units.

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173. For what value of k are the points (1, 1), (3, k) and (-1, 4) collinear.

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174. For what value of p are the points (2, 1), (p, -1) and (-1, 3) collinear.

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175. Find the area of the rhombus whose vertices taken in order are the points (3, 0), (4, 5), (-1, 4) and (-2, -1).

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176. Find the distance between the points
(cos ө - sin ө), (-cos ө, sin ө)

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177. Find the distance between the points
Let the given points be A (a cos ө, 0) and B(0, a sin ө)

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178. Find the distance between the points
Let the given points be A(0, 0) and B(6, 8) we know that distance between two points A(x₁, y₁) and B(x₂, y₂) is given by

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179. Find the co-ordinates of the mid-point of the line segment joining (-5, 4) and (6, -7).

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180. Find the co-ordinates of the mid-point of the line segment joining (4, 7) and (6, 9).

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