CBSE Class 10 Coordinate geometry Sure Shot Questions Set 09

Question. Draw a triangle PQR whose vertices are P = \( (1, – 6) \), Q = \( (7, 4) \) and R = \( (– 4, 4) \).
Answer: Triangle PQR plotted with base QR on line \( y = 4 \) with length 11 units.

Question. Draw a triangle ABC whose vertices A, B, and C are \( (– 3, 0) \), \( (3, 3) \) and \( (– 3, 3) \) respectively.
Answer: Right-angled triangle plotted with right angle at \( (–3, 3) \).

Question. Draw a rectangle ABCD such that its vertices A, B, C and D are \( (4, 3) \), \( (4, – 2) \), \( (– 7, – 2) \) and \( (– 7, 3) \) respectively.
Answer: Rectangle plotted with length 11 units and breadth 5 units.

Question. Draw a rectangle KLMN such that its vertices K, L, M, and N are \( (5, 0) \), \( (5, 3) \), \( (0, 3) \) and \( (0, 0) \) respectively.
Answer: Rectangle plotted with one vertex at the origin, length 5 and height 3.

Question. Construct a square ABCD such that its vertices A, B, C, and D are \( (1, 2,) \) \( (– 7, 2) \), \( (– 7, – 6) \) and \( (1, – 6) \) respectively.
Answer: Square plotted with side length 8 units.

Question. Construct a square PQRS whose vertices P, Q, R and S are \( (0, 0) \), \( (– 4, 0) \), \( (– 4 , – 4) \) and \( (0, – 4) \) respectively
Answer: Square plotted in Quadrant III with side length 4 units.

Question. Draw a parallelogram ABCD whose vertices A, B, C, and D are \( (– 4, 8) \), \( (– 4, 2) \), \( (6, – 7) \) and \( (6, – 1) \) respectively.
Answer: Parallelogram plotted with vertical sides of length 6 units.

Question. Construct a trapezium PQRS in which vertices P, Q, R and S are \( (3, 0) \), \( (7, 9) \), \( (– 6, 9) \) and \( (– 2, 0) \) respectively.
Answer: Trapezium plotted with parallel bases PQ and RS on \( y = 0 \) and \( y = 9 \).

Question. Draw a rhombus ABCD whose vertices A, B, C and D are \( (1, 4.5) \), \( (– 1, 0) \), \( (1, – 4.5) \) and \( (3, 0) \) respectively
Answer: Rhombus plotted with diagonals of length 9 and 4 units.

Question. Find the distance between the following pair of points :
(i) \( (– 6, 7) \) and \( (– 1, – 5) \)
(ii) \( (a + b, b + c) \) and \( (a – b, c – b) \)
(iii) \( (a \sin \alpha, – b \cos \alpha) \) and \( (– a \cos \alpha, b \sin \alpha) \)
(iv) \( (a, 0) \) and \( (0, b) \)
Answer: (i) 13
(ii) \( 2b\sqrt{2} \)
(iii) \( \sqrt{a^2+b^2}(\sin\alpha + \cos\alpha) \)
(iv) \( \sqrt{a^2+b^2} \)

Question. Find the value of a when the distance between the points \( (3, a) \) and \( (4, 1) \) is \( \sqrt{10} \).
Answer: 4, –2

Question. Which point on x-axis is equidistant from \( (5, 9) \) and \( (– 4, 6) \) ?
Answer: \( (3, 0) \)

Question. Three vertices of a parallelogram are \( (a + b, a – b) \), \( (2a + b, 2a – b) \), \( (a – b, a + b) \), find the fourth vertex.
Answer: \( (–b, b) \)

Question. If the coordinates of the mid-points of the sides of a triangle are \( (1, 1) \), \( (2, – 3) \) and \( (3, 4) \), find its vertices.
Answer: \( (2, 10), (0, – 4), (4, 2) \)

Question. Find the centroid of the triangle whose vertices are :
(i) \( (1, 4) \), \( (– 1, –1) \), \( (3, – 2) \)
(ii) \( (– 2, 3) \), \( (2, –1) \), \( (4, 0) \)
Answer: (i) \( (1, 1/3) \), (ii) \( (4/3, 2/3) \)

Question. Two vertices of a triangle are \( (1, 2) \), \( (3, 5) \) and its centroid is at the origin. Find the coordinates of the third vertex.
Answer: \( (–4, –7) \)

Question. If \( (– 2, 3) \), \( (4, – 3) \) and \( (4, 5) \) are the mid-points of the sides of a triangle, find the coordinates of its centroid.
Answer: \( (2, 5/3) \)

Question. Find the area of a triangle whose vertices are
(i) \( (6, 3) \), \( (– 3, 5) \) and \( (4, – 2) \)
(ii) \( (at_1^2, 2at_1) \), \( (at_2^2, 2at_2) \) and \( (at_3^2, 2at_3) \)
(iii) \( (a, c + a) \), \( (a, c) \) and \( (– a, c – a) \)
Answer: (i) 49/2 sq. units
(ii) \( a^2(t_1 - t_2)(t_2 - t_3)(t_1 - t_3) \)
(iii) \( a^2 \)

Question. Prove that \( (2, – 2) \), \( (– 2, 1) \) and \( (5, 2) \) are the vertices of a right angled triangle. Find the area of the triangle and the length of the hypotenuse.
Answer: Area = 12.5 sq. units, Hypotenuse = \( \sqrt{50} \) or \( \sqrt{65} \)

Question. Prove that the points \( (2a, 4a) \), \( (2a, 6a) \) and \( (2a + \sqrt{3} a, 5a) \) are the vertices of an equilateral triangle.
Answer: Proved by checking all side lengths are equal to \( 2a \).

Question. Prove that the points \( (2, 3) \), \( (– 4, – 6) \) and \( (1, 3/2) \) do not form a triangle
Answer: Proved as area of triangle is 0 (points are collinear).

Question. An equilateral triangle has two vertices at the points \( (3, 4) \) and \( (– 2, 3) \), find the coordinates of the third vertex.
Answer: \( \left( \frac{1+5\sqrt{3}}{2}, \frac{7-5\sqrt{3}}{2} \right) \) or \( \left( \frac{1-5\sqrt{3}}{2}, \frac{7+5\sqrt{3}}{2} \right) \)

Question. Two vertices of an isosceles triangle are \( (2, 0) \) and \( (2, 5) \). Find the third vertex if the length of the equal sides is 3.
Answer: \( \left( 2 - \frac{\sqrt{11}}{2}, 2.5 \right) \) or \( \left( 2 + \frac{\sqrt{11}}{2}, 2.5 \right) \)

Question. Find the value of k, if the point P \( (0, 2) \) is equidistant from \( (3, k) \) and \( (k, 5) \).
Answer: 1

Question. Find the coordinates of the point which divides the line segment joining \( (– 1, 3) \), and \( (4, – 7) \) internally in the ratio. 3 : 4.
Answer: \( (8/7, –9/7) \)

Question. Find the point of trisection of the line segment joining the points :
(i) \( (5, – 6) \) and \( (– 7, 5) \)
(ii) \( (3, – 2) \) and \( (– 3, – 4) \)
(iii) \( (1, 2) \) and \( (11, 9) \).
Answer: (i) \( (1, –7/3), (–3, 4/3) \)
(ii) \( (1, –8/3), (–1, –10/3) \)
(iii) \( (13/3, 13/3), (23/3, 20/3) \)

Question. Three consecutive vertices of a parallelogram are \( (–2, –1) \), \( (1, 0) \) and \( (4, 3) \). Find the fourth vertex.
Answer: \( (1, 2) \)

Question. If A \( (–1, 3) \), B \( (1, –1) \) and C \( (5, 1) \) are the vertices of a triangle ABC, find the length of the median through A.
Answer: 5

Question. If the coordinates of the mid-points of the sides of a triangle are \( (1, 1) \), \( (2, –3) \) and \( (3, 4) \), find the vertices of the triangle.
Answer: \( (4, 0), (2, 8), (0, –6) \)

Question. If the mid-point of the line joining \( (3, 4) \) and \( (k, 7) \) is \( (x, y) \) and \( 2x + 2y + 1 = 0 \) find the value of k.
Answer: k = – 15

Question. Determine the ratio in which the straight line \( x – y – 2 = 0 \) divides the line segment joining \( (3, –1) \) and \( (8, 9) \).
Answer: 2 : 3 Internally

Question. Determine the ratio in which the point P \( (m, 6) \) divides the join of A \( (–4, 3) \) and B \( (2, 8) \). Also find the value of m.
Answer: 3 : 2, m = – 2/5

Question. Determine the ratio in which the point \( (– 6, a) \) divides the join of A \( (– 3, 1) \) and B \( (– 8, 9) \). Also find the value of a.
Answer: 3 : 2, a = 29/5

Question. Find the area of the quadrilaterals, the coordinates of whose vertices are
(i) \( (– 3, 2) \), \( (5, 4) \), \( (7, – 6) \) and \( (– 5, – 4) \)
(ii) \( (1, 2) \), \( (6, 2) \), \( (5, 3) \) and \( (3, 4) \)
Answer: (i) 80 sq. units, (ii) 11/2 sq. units

Question. The four vertices of a quadrilateral are \( (1, 2) \), \( (–5, 6) \), \( (7, –4) \) and \( (k, –2) \) taken in order. If the area of the quadrilateral is zero, find the value of k.
Answer: k = 3

Question. Find the centre of the circle passing through \( (5, – 8) \), \( (2, – 9) \) and \( (2, 1) \).
Answer: \( (2, –4) \)

Question. Find the value of x such that PQ = QR where the coordinates of P, Q and R are \( (6, – 1) \), \( (1, 3) \) and \( (x, 8) \) respectively.
Answer: 5, –3

Question. Find the centre of the circle passing through \( (6, –6) \), \( (3, –7) \) and \( (3, 3) \).
Answer: \( (3 , –2) \)

Question. Two opposite vertices of square are \( (–1, 2) \) and \( (3, 2) \). Find the coordinates of other two vertices.
Answer: \( (1, 0) \) and \( (1, 4) \)

Question. The area of a triangle is 5. Two of its vertices are \( (2, 1) \) and \( (3, –2) \). The third vertex lies on \( y = x + 3 \). Find the third vertex.
Answer: \( (7/2, 13/2) \) or \( (–3/2, 3/2) \)

Question. Four points A \( (6, 3) \), B \( (– 3, 5) \), C \( (4, – 2) \), and D \( (x, 3x) \) are given in such a way that \( \frac{\Delta DBC}{\Delta ABC} = \frac{1}{2} \), find x.
Answer: 11/8

Question. For what value of a the point \( (a, 1), (1, – 1) \) and \( (11, 4) \) are collinear ?
Answer: a = 5