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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).

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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).

We know that, ar (∆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

Here, we have
x₁ = 3,    y₁ = 0
x₂ = 4,    y₂ = 5
and    x₃ = -1,    y₃ = 4
Now, area of ∆ABD

$equals 1 half left square bracket 3 left parenthesis 5 minus 4 right parenthesis plus 4 left parenthesis 4 minus 0 right parenthesis plus left parenthesis negative 1 right parenthesis left parenthesis 0 minus 5 right parenthesis right square bracket equals 1 half left square bracket 3 space straight x space 1 space plus 4 space straight x space 4 space plus space left parenthesis negative 1 right parenthesis space left parenthesis negative 5 right parenthesis right square bracket equals 1 half left square bracket 3 plus 16 plus 5 right square bracket equals 1 half straight x space 24 space equals space 128 space s q space u n i t s.$

Here,
Now, area of rhombus (ABCD)
= 2 × ar (∆ABC)
2 × 12 = 24 Sq. units.

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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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