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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.

146 Views

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.

2655 Views

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.

298 Views

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

236 Views

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?

Let A(2, -1), B(3, 4), C(-2, 3) and D(x, y) be the vertices of a rhombus.

Fig. 7.32.
Co-ordinates of the mid-point of AC are :

$equals open square brackets fraction numerator straight x subscript 1 plus straight x subscript 2 over denominator 2 end fraction comma space fraction numerator straight y subscript 1 plus straight y subscript 2 over denominator 2 end fraction close square brackets equals open square brackets fraction numerator 2 plus left parenthesis negative 2 right parenthesis over denominator 2 end fraction comma space fraction numerator negative 1 plus 3 over denominator 2 end fraction close square brackets equals open square brackets fraction numerator 2 minus 2 over denominator 2 end fraction comma space fraction numerator negative 1 plus 3 over denominator 2 end fraction close square brackets equals open square brackets 0 over 2 comma space 2 over 2 close square brackets equals left square bracket 0 comma 1 right square bracket$
Co-ordinates of the mid-point of BD are

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We know that the diagonals of rhombus bisect each other.
Therefore, co-ordinates of mid-point of AC = co-ordinates of mid-point of BD

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Hence, the co-ordinates of the fourth vertex is (-3, -2).
Case II. Test for square :

$AC space equals space square root of left parenthesis negative 2 minus 2 right parenthesis squared plus left curly bracket 3 minus left parenthesis negative 1 right parenthesis right curly bracket squared end root space space space space space space equals space square root of left parenthesis negative 4 right parenthesis squared plus left parenthesis 3 plus 1 right parenthesis squared end root space space space space space space space equals square root of 16 plus 16 end root equals square root of 32 space space space space space space space equals space 4 square root of 2$

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Clearly, AC ≠ BD.
So, ABCD is not a square.

1151 Views

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.

89 Views

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

2113 Views

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

436 Views

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.

187 Views

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

2329 Views