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Coordinate Geometry

Short Answer Type

211. Find (he ratio in which the y-axis divides the line segment joining the points (5, -6) and (-1, -4). Also find the point of intersection.

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212. Find the ratio in which the line segment joining the points (6, 4) and (1, - 7) is divided by x-axis. Also find the point of intersection.

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213. If the points A(6, 1), B(8, 2), C(9, 4) and D(p, 3) are the vertices of a parallelogram taken in order, find the value of p.

The mid-point of the diagonal AC is

equals space open square brackets fraction numerator straight x subscript 1 plus straight x subscript 2 over denominator 2 end fraction comma space fraction numerator straight y subscript 1 plus straight y subscript 2 over denominator 2 end fraction close square brackets

[Here, we have x₁ = 6, y₁ = 1
x₂ = 9, y₂ = 4]

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and the mid-point of the diagonal BD is

$equals open square brackets fraction numerator straight x subscript 1 plus straight x subscript 2 over denominator 2 end fraction comma space fraction numerator straight y subscript 1 plus straight y subscript 2 over denominator 2 end fraction close square brackets$

Here, we have x₁ = 8, y₁ = 2
x₂ = p, y₂ = 3]

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Since, the co-ordinates of the mid-point of AC = Co-ordinates of the mid point of BD.

$rightwards double arrow space open square brackets 15 over 2 comma space 5 over 2 close square brackets equals open square brackets fraction numerator 8 plus straight p over denominator 2 end fraction comma space 5 over 2 close square brackets rightwards double arrow space fraction numerator 8 plus straight p over denominator 2 end fraction equals 15 over 2 rightwards double arrow space space 8 plus straight p equals 15 rightwards double arrow space straight p space equals space 15 space minus 8 rightwards double arrow space straight p space equals space 7$

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214. Prove (hat the points (-2, -1), (1, 0), (4, 3) and (1, 2) are the vertices of a parallelogram is it a rectangle?

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215. The three vertices of a rhombus, taken in order are (2, -1), (3, 4) and (-2, 3). Find the fourth vertex, is it a square?

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216. The mid-points of the sides of a triangle are (3, 4), (4, 1) and (2, 0). Find the coordinates of the vertices of the triangle.

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217. Find the lengths of the medians of the triangle whose vertices are (1, -1), (0, 4) and (-5, 3).

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218. Determine the ratio in which the line 3x + y - 9 = 0 divides the segment joining the points (1, 3) and (2, 7).

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219. 26. If C is a point lying on the line segment AB joining A(1, 1) and B(2, - 3) seen that 3AC = CB, then find the co-ordinates of C.

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220. Prove that the mid-point of the hypotenuse of a right angled triangle is equidistant from its vertices.

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