# Coordinate Geometry

**Exercise 1**

**1.** Find the distance between the following pairs of points:

(i) (2, 3), (4,1)

(ii) (-5, 7), (-1, 3)

(iii) (a, b), (-a, -b)

**2.** Find the distance between the points (0, 0) and (36, 15).Also, find the distance between towns A and B if town B is located at 36 km east and15 km north of town A.

**3.** Determine if the points (1, 5), (2, 3) and (-2, -11) are collinear.

**4.** Check whether (5, -2), (6, 4) and (7, -2) are the vertices of an isosceles triangle.

**5. ** In a classroom, 4 friends are seated at the points A (3, 4), B (6, 7), C (9, 4) and D (6, 1). Champa and Chameli walk into the class and after observing for a few minutes Champa asks Chameli. “Don’t you think ABCD is a square?” Chameli disagrees. Using distance formula, find which of them is correct.

**6.**Name the type of quadrilateral formed, if any, by the following points, and give reasons for your answer.

(i) (-1, -2), (1, 0), (-1, 2), (-3, 0)

(ii) (-3, 5), (3, 1), (0, 3) , (-1, -4)

(iii) (4, 5), (7, 6), (4, 3), (1, 2)

**7.** Find the point on the x-axis which is equidistant from (2, -5) and (-2, 9).

**8.** Find the values of y for which the distance between the points P (2, -3) and Q (10, y) is 10 units.

**9.** If, Q (0, 1) is equidistant from P (5, -3) and R (x, 6), find the values of x. Also, find the distances QR and PR.

**10.** Find a relation between x and y such that the point (x, y)is equidistant from the point (3, 6) and (-3, 4).

**Exercise 2**

**11.** Find the coordinates of the point which divides the join of (-1, 7) and (4, -3) in the ratio 2:3.

**12.** Find the coordinates of the points of trisection of the line segment joining (4, -1) and (-2, -3).

**13.** To conduct sports day activities, in your rectangular shaped school ground ABCD, lines have been drawn with chalk powder at a distance of 1 m each. 100 flower pots have been placed at a distance of 1 m from each other along AD. Niharika runs 14th of the distance AD on the 2nd line and posts a green flag. Preet runs 15th of the distance AD on the eighth line and posts a red flag. What is the distance between both the flags? If Rashmi has to post a blue flag exactly halfway between the line segment joining the two flags, where should she post her flag?

**14.** Find the ratio in which the line segment joining the points (-3, 10) and (6, -8) is divided by (-1, 6).

**15.** Find the ratio in which the line segment joining A (1, -5) and B (-4, 5) is divided by the x-axis. Also find the coordinates of the point of division.

**16.** If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find x and y.

**17.** Find the coordinates of a point A, where AB is the diameter of a circle whose centre is (2, -3) and B is (1, 4).

**18.** If A and B are (-2, -2) and (2, -4) respectively, find the coordinates of P such that AP = AB and P lies on the line segment AB.

**19.** Find the coordinates of the points which divides the line segment joining A (-2, 2) and B (2, 8) into four equal parts.

**20.** Find the area of a rhombus if its vertices are (3, 0), (4, 5), (-1, 4) and (-2, -1) taken in order. **{Hint: Area of a rhombus = 1/2 (product of its diagonals))**

**Exercise 3**

**21.** Find the area of the triangle whose vertices are:

(i) (2, 3), (-1, 0), (2, -4)

(ii) (-5, -1), (3, -5), (5, 2)

**22.** In each of the following find the value of ‘k’, for which the points are collinear.

(i) (7, -2), (5, 1), (3, k)

(ii) (8, 1), (k, -4), (2, -5)

**23.** 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 this area to the area of the given triangle.

**24.** Find the area of the quadrilateral whose vertices taken in order are (-4, -2), (-3, -5), (3, -2) and (2, 3).

**25.** We know that median of a triangle divides it into two triangles of equal areas. Verify this result for ΔABC whose vertices are A (4, -6), B (3, -2) and C (5, 2).

**Exercise 4**

**26.** Determine the ratio in which the line 2x + y – 4 = 0 divides the line segment joining the points A (2,-2) and B (3, 7).

**27.** Find a relation between x and y if the points (x, y), (1, 2) and (7, 0) are collinear.

**28.** Find the centre of a circle passing through the points (6, -6), (3, -7) and (3, 3).

**29.** The two opposite vertices of a square are (-1, 2) and (3, 2). Find the coordinates of the other two vertices.

**30.** The class X students of a secondary school in Krishinagar have been allotted a rectangular plot of land for their gardening activity. Saplings of Gulmohar are planted on the boundary at a distance of 1 m from each other. There is a triangular grassy lawn in the plot as shown in the figure. The students are to sow seeds of flowering plants on the remaining area of the plot.

(i) Taking A as origin, find the coordinates of the vertices of the triangle.

(ii) What will be the coordinates of the vertices of Δ PQR if C is the origin? Also calculate the area of the triangle in these cases. What do you observe?

**31.**The vertices of a Δ 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 . Calculate the area of the Δ ADE and compare it with the area of Δ ABC.

**32.** Let A (4, 2), B (6, 5) and C (1, 4) be the vertices of Δ ABC.

(i) The median from A meets BC at D. Find the coordinates of the point D.

(ii) Find the coordinates of the point P on AD such that AP : PD = 2 : 1.

(iii) Find the coordinates of points Q and R on medians BE and CF respectively such that BQ : QE = 2 : 1 and CR : RF = 2 : 1.

(iv) What do you observe?

(Note: The point which is common to all the three medians is called *centroid *and this point divides each median in the ratio 2 : 1)

(v) If A (x_{1}, y_{1}), B (x_{2}, y_{2}) and C (x_{3}, y_{3}) are the vertices of Δ ABC, find the coordinates of the centroid of the triangle.

**33.** ABCD is a rectangle formed by joining points A (-1, -1), B (-1, 4), C (5, 4) and D (5, -1). P, Q, R and S are the mid-points of AB, BC, CD and DA respectively. Is the quadrilateral PQRS a square? Or a rhombus? Justify your answer.