Section I 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)
2. Determine if the points (1, 5), (2, 3) and (2,11) are collinear.
3. Check whether (5, 2),(6, 4) and (7, 2) are the vertices of an isosceles triangle.
4. In a classroom, 4 friends are seated at the points A, B, C and D as shown in the following figure. 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.
5. 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)
6. Find the point on the xaxis which is equidistant from (2, 5) and ( 2, 9).
7. Find the values of y for which the distance between the points P (2, 3) and Q (10, y) is 10 units.
8. If Q (0, 1) is equidistant from P (5, 3) and R (x, 6), find the values of x. Also find the distance QR and PR.
9. Find a relation between x and y such that the point (x, y) is equidistant from the point (3, 6) and ( 3, 4).
Section II Question:
10.Find the coordinates of the point which divides the join of (1, 7) and (4, 3 ) in the ratio 2:3.
11.Find the coordinates of the points of trisection of the line segment joining (4, 1) and (2, 3).
12.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, as shown in the following figure. Niharika runs 1/4th the distance AD on the 2nd line and posts a green flag. Preet runs 1/5th 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 flagexactly halfway between the line segment joining the two flags, where should she post her flag?
13. Find the ratio in which the line segment joining the points (3, 10) and (6, 8) is divided by (1, 6).
14.Find the ratio in which the line segment joining A (1, 5) and B ( 4, 5) is divided by the xaxis. Also find the coordinates of the point of division.
15.If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find x and y.
16.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).
17.If A and B are (– 2, – 2) and (2, – 4), respectively, find the coordinates of P such that AP = 3 /7AB and P lies on the line segment AB.
18.Find the coordinates of the points which divide the line segment joining A (– 2, 2) and B(2, 8) into four equal parts.
19.Find the area of a rhombus if its vertices are (3, 0), (4, 5), (– 1, 4) and (– 2, – 1) taken in order.
Sections III Question:
20.Find the area of the triangle whose vertices are :
(i) (2, 3), (–1, 0), (2, – 4) (ii) (–5, –1), (3, –5), (5, 2)
21.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)
22.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.
23.Find the area of the quadrilateral whose vertices, taken in order, are (– 4, – 2), (– 3, – 5), (3, – 2) and (2, 3).
24.You have studied in Class IX that a 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).
Miscellaneous:
1. Show that points (a, b + c), (b, c +a) and (c, a+ b) are collinear.
2. Triangle ABC is given, Such that co-ordinates of vertex A= (x1, y1) , B (x2, y2) , C= (x3,y3).Then find the co-ordinate of centroid of triangle ABC.
Fill in the Blanks:
1. Point (- 3,0) lies on ——-
2. Point (0, 7) lies on___________.
3. Coordinates of origin are___________.
4. Distances of points (3,4) and (4,3) from origin are___________and is equal to
5. Area of triangle whose vertices are (0,0) (3,0) and (0,4) is___________.
6. Distance of point whose coordinates are (x, y) from x-axis is ___________.
7. Distance of point whose coordinates are (a, b) from y-axis is___________.
8. Image of point (2, - 3) under x-axis is___________.
9. Image of point (- 7,5) under y-axis is___________.
10. Image of point (0,5) under y-axis is___________.
11. Image of point (-4,0) under x-axis is___________.
State True or False:
1. Ordinate on x-axis is always zero.
2. Abscissa on x-axis is always negative.
3. xcoordinate on y-axis is always zero.
4. Coordinates of centroid of a formed by joining mid-points is same as of the original triangle.
5. Trisection of a line segment means ratio of 1: 2 or 2 : 1.
6. Point P (- 4.2) lies on the line segment joining the points A (- 4, 6) and B (- 4, - 6).
7. The point (0,5), (0, - 9) and (3.6) are collinear.
8. Point P (0, 2) is the point of intersection of y-axis and perpendicular bisector of line segment joining the points A (- 1.1) and B (3.3).
9. Points A (3, 1), B (12, - 2) and C (0,2) cannot be the vertices of a triangle.
10. Points A (4, 3), B (6,4), C (5, - 6) and D (- 3,5) are the vertices of a parallelogram.
11. A circle has its centre at the origin and a point P (3,0) lies on it. The point Q (6, 8) lies outside the circle.
12. The point A (2, 7) lies on the perpendicular bisector of line segment joining the points P (6, 5) and Q (0, - 4).
13. Point P (5, - 3) is one of the two points of trisection of the line segment joining the points A (7, - 2) and B (1, - 5).
14. Point A (- 6,10), B (- 4,6) and C (3, - 8) are collinear such that AB = 2/9*AC.
15. The point P (- 2,4) lies on a circle of radius 6 and centre C (3,5).