Find the distance between the following pairs of points:

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

(ii) (– 5, 7), (– 1, 3)

(iii) (a, b), (– a, – b)

Find the distance between the points (0, 0) and (36, 15). Can you now find the distancebetween the two towns A and B discussed in Section 7.2

Determine if the points (1, 5), (2, 3) and (– 2, – 11) are collinear

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

In a classroom, 4 friends areseated at the points A, B, C and D as shown in Fig. 7.8 Champaand Chameli walk into the classand after observing for a fewminutes Champa asks Chameli,“Don’t you think ABCD is asquare?” Chameli disagrees.Using distance formula, findwhich of them is correct

Name the type of quadrilateralformed, if any, by the followingpoints, and give reasons foryour 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)

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

Find the values of y for which the distance between the points P(2, – 3) and Q (10, y) is10 units

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

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

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

Find the coordinates of the points of trisection of the line segment joining (4,–1) and (–2, –3)

To conduct Sports Day activities, in your rectangular shaped school ground ABCD, lines have been drawn with chalk powder at a distance of 1m each. 100 flower pots have been placed at a distance of 1m from each other along AD, as shown in Fig. 7.12. Niharika runs th thedistance AD on the 2nd line andposts a green flag. Preet runsththe distance AD on the eighth lineand posts a red flag. What is thedistance between both the flags? IfRashmi has to post a blue flag exactlyhalfway between the line segmentjoining the two flags, where shouldshe post her flag?

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

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

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

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)

If A and B are (– 2, – 2) and (2, – 4), respectively, find the coordinates of P such thatand P lies on the line segment AB

Find the coordinates of the points which divide the line segment joining A(– 2, 2) andB (2, 8) into four equal parts

Find the area of a rhombus if its vertices are (3, 0), (4, 5), (– 1, 4) and (– 2, – 1) taken inorder. [Hint: Area of a rhombus (product of its diagonals)]

Find the area of the triangle whose vertices are:

(i) (2, 3), (–1, 0), (2, – 4)

(ii) (–5, –1), (3, –5), (5, 2)

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)

Find the area of the triangle formed by joining the mid-points of the sides of the trianglewhose vertices are (0, –1), (2, 1) and (0, 3). Find the ratio of this area to the area of the given triangle

Find the area of the quadrilateral whose vertices, taken in order, are (– 4, – 2),(–3, – 5), (3, – 2) and (2, 3)

You have studied in Class IX, (Chapter 9, Example 3), that a median of a triangle dividesit into two triangles of equal areas. Verify this result for Δ ABC whose vertices areA(4, – 6), B(3, –2) and C (5, 2)

Determine the ratio in which the line divides the line segment joining the points A (2, – 2) and B (3, 7)

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

Find the centre of a circle passing through the points (6, – 6), (3, – 7) and (3, 3)

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

The Class X students of asecondary school inKrishinagarhave been allotteda rectangular plot of land fortheir gardening activity.Sapling of Gulmohar areplanted on the boundary at adistance of 1m from each other.There is a triangular grassylawn in the plot as shown inthe Fig. 7.14. The students areto sow seeds of floweringplants 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 areas of the triangles in these cases. What do you observe?

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 theΔ ADE and compare it with the area of Δ ABC (Recall Theorem 6.2 and Theorem 6.6)

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 the centroidand this point divides each median in the ratio 2: 1]

(v) If and are the vertices of Δ ABC, find the coordinates of the centroid of the triangle

ABCD is a rectangle formed by the 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 quadrilateralPQRS a square? a rectangle? or a rhombus? Justify your answer