#### Exercise 7.1

**Question 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)**

**Note : Distance between the two points can never be negative.**

**Question 2. Find the distance between the points (0, 0) and (36, 15). Can you find the distance between the two towns A and B discussed. A town B is located 36 km east and 15 km north of the town A.**

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

**Note: Three points A, Band Care collinear or lie on a line if one of the following holds(i) AB+BC= AC (ii) AC+ CB= AB (iii) CA + AB = CB.**

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

Question 5. In a classroom, 4 friends are seated at the points A, B, C and D as shown in Fig. Champa and Chameli walk into the class and after observing for a few minutes Champa asks chameli, “Don’t you think ABCD is a rectangle?” Chameli disagrees. Using distance formula find which of them is correct, any why?

Note : Every square is a rectangle, but not all rectangles are squares.

Question 6. Name the type of quadrilateral formed, if any, by the following points, and give reasons for your answers:

(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)

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

**Note :** The ordinate of the point on x-axis is 0

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

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

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

### Exercise 7.2

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

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

Note : Since Q is the mid point of PB, it can also be obtained using mid-point formula

**Question 3. To conduct Sports Day activities, in your rectangular shaped school ground ABCD, lines have been drawn with chalk powder at a distance of l m each.100 flower pots have been placed at a distance of lm from each other along AD, as shown in Fig. 7.12. Niharika runs 1/4 th the distance AD on the 2nd line and posts a green flag. Preet runs the 1/5th the distance AD on the eighth line and posts a red flag. What is the distance between both the flags? IfRashmi has to post a blue flag exactly half-way between the line segment joining the two flags, where should she post her flag?**

**Question 4. Find the ratio in which the line segment joining the points (- 3, 10) and (6, – 8) is divided by (- 1, 6).**

**Question 5. 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.**

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

**Question 7. 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).**

**Question 8. If A and B are (- 2, – 2) and (2, – 4), respectively, find the coordinates of P such that AP = 3/7 AB and P lies on the line segment AB.**

**Question 9. Find the coordinates of the points which divide the line segment joining A(- 2, 2) and B(2, 8) into four equal parts.**

Question 10. 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 = (product of its diagonals)]

#### Exercise 7.3

**Question 1. Find the area of the triangle whose(i) (2,3), (-1, 0), (2,-4) (ii) (-5,-1), (3,-5), (5, 2)**

**Note :** Since area is measure, it cannot be negative.

**Question 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)**

**Note : Collinearity of three points can be proven using any of the given conditions.(i) If the sum of the lengths of any two line segments among AB, BC and AC is equal to the length of the remaining line segment, i.e.,AB +BC= CA, or AB+ AC =BC, or AC+ BC =AB.(ii) If area of MBC = 0,**

**Question 3. 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.**

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

Note : To find the area of any polygon, simply divide it into triangular regions having no common area, and then add the areas of these regions.

**Question 5. You have studied in Class IX (Chapter 9, example 3) that a median of a triangle divides it into two triangles of equal areas. Verify this result for ABC whose vertices are A(4, – 6), B(3, -2) and C(5, 2).**

#### Exercise 7.4

**Question 1. Determine the ratioin which the line 2x+y-4 =0 divides the line segment joining the points A(2,- 2) and B(3, 7).**

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

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

**Question 4. The two opposite vertices of a square are (-1, 2) and (3, 2). Find the coordinates of the other two vertices.**

**Question 5. The Class X students of a secondary school in Krishinagar have been allotted a rectangular plot ofland for their gardening activity. Sapling of Gulmohar are planted on the boundary at a distance of lm from each other. There is a triangular grassy lawn in the plot as shown in the Fig. 7.14. 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 citheva-ticesci.:PQR if C is the origin? Also calculate the areas of the triangles in these cases. What do you observe?**

Question 6. The vertices of aMBC 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 the MOE and compare it with the area of ABC.

**Question 7. Let A(4, 2), B(6, 5) and C(l, 4) be the vertices of AABC.(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 centroid and this point divides each median in the ratio 2 : 1.](v) If A(x,,y,), B(x2,y2) and C(x”y,) are the vertices of ABC, find the coordinates of the centroid of the triangle.**

**Question 8. ABCD is a rectangle formed by the points A(-1,-1), B(- 1,4), C(S,4) and D(5,-1).P, Q, Rand Sare the mid-points of AB, BC, CD and DA, respectively. Is the quadrilateral PQRS a square? a rectangle? or a rhombus? Justify your answer.**

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