Q. Find the coordinates of the centroid of a triangle.
- Sol. Let the vertices of ∆ABC be A = (x_{1}, y_{1}), B = (x_{2}, y_{2}) and C = (x_{3}, y_{3}). Also, let the vertices of the centroid be G = (x, y).
Q. Find the point on the X-axis which is equidistant from the points A(2, – 5) and B(– 2, 9).
Ans. (– 7, 0).
Q. Find the point on the Y-axis which is equidistant from the points (6, 5) and (– 4, 3).
Ans. (0, 9).
Q. If the point Q(0, 1) is equidistant from the points P(5, – 3) and R(x, 6), find the values of x. Also find the distances QR and PR.
- Ans. x = ±4, QR = √41, PR = √82 or √92 units.
Q. Find the values of y for which the distance between the points P(2, – 3) and Q(10, y) is 10 units.
- Ans. 3 or – 9.
Q. Name the quadrilateral formed by the following points A(4, 5), B(7, 6), C(4, 3) and D(1, 2).
- Ans. Parallelogram.
Q. Find the equation of the perpendicular bisector of the line segment joining the points (7, 1) and (3, 5).
- Ans. x – y = 2.
Q. Find the relation between x and y such that the point (x, y) is equidistant from the points (3, 6) and (– 3, 4).
- Ans. 3x + y = 5.
Q. Find the coordinates of the points of trisection of the line segment joining the points A(2, – 2) and B(– 7, 4).
- Ans. (– 1, 0) and (– 4, 2)
Q. If A and B are two points having coordinates (– 2, – 2) and (2, – 4) respectively, find the coordinates of P such that AP = 3/7 AB
- Ans. (-2/7, 20/7).
Q. Find the coordinates of the points which divide the line segment joining A(– 2, 2) and B(2, 8) into four equal parts.
- Ans. (-1,7/2),(0, 5), (1, 13/2).
Q. ABCD is a rectangle where A(– 1, – 1), B(– 1, 4), C(5, 4) and D(5, – 1) are the vertices. P, Q, R, S are the mid-points of sides AB, BC, CD and AD respectively. Is the quadrilateral PQRS a square or a rectangle or a rhombus ? Justify your answer.
- Ans. Rhombus because all the sides of quadrilateral are equal and diagonals are not equal to each other.
Q. Find the ratio in which the point P(– 4, 6) divides the line segment joining the points A(– 6, 10) and B(3, – 8).
- Ans. 2 : 7.
Q. Find the coordinates of a point A where AB is a diameter of a circle whose centre is (2, – 3) and B is (1, 4).
- Ans. (3, – 10).
Q. Find the area of the triangle formed by joining the mid-points of the sides of the triangle whose vertices are (0, – 1), (2, 1) and (0, 3). Find the ratio of the area of a triangle formed to the area of the given triangle.
- Ans. 1 sq. units, 1 : 4.
Q. The vertices of ∆ABC are A = (4, 6), B = (1, 5) and C = (7, 2). A line is drawn to intersect sides AB and AC at D and E respectively such that AD/AB= AE/AC=1/4. Calculate the area of ∆ADE and compare it with the area of ∆ABC.
- Ans. 15/32 sq. units, 1 : 16.
Q. Find the area of the quadrilateral ABCD whose vertices are A(– 4, – 2), B(– 3, – 5), C(3, – 2) and D(2, 3).
- Ans. 28 sq. units.