Fill in the blanks:
1) If a point lies on x-axis, then y coordinates is ______
2) If a point lies on y-axis, then x coordinates is ______
3) If two points have same absicca, the line joining them is parallel to ____
4) If two points have same ordinate, the line joining them is parallel to ____
5) The line joining (2,3) and (-2,-3) passes through the ____.
6) The distance along the x-axis is called ___
7) The distance along the y-axis is called ___
8) XOX' is called the ____
9) YOY' is called the _____
EXERCISE -B
1) Find the distance between the points
A) P(-2,3), Q(6,-3). 10
B) A(2,3), B(-2,3). 4
C) X(3,5), Y(3,1). 4
D) P(2,-), Q(5,2). 5
E) M(3,7), N(-2,-5). 13
F) P(2/5,2/5), Q(-2/5,-1/5). 1
G) P(5/10,1/10), Q(-1/10,-7/10). 1
H) P(√3,1), Q(0,0). 2
I) X(√2+1, 2), Y(1, 2 - √2). 2
J) (a,b),(-a,-b). 2√(a²+b²)
2) Show that (3,3) is the centre of the circle passing through the points (6,2) (0,4) and (4,6).
EXERCISE -C
1)a) If A(4,2) and B(1,y), find the possible values of y so that AB= 5. -2,.6
b) If A(2,-3) and B(10,y), find the possible values of y so that AB= 10. 3,9
c) If A(2,5) and B(x,-7), find the possible values of x so that AB= 13. -3, 7
d) Find the value of a, if the distance between the points A(-3,-14) and B(a,-5) is 9 units. -3
e) A line segment is of length 10 units and one of its end is (-2,3). If the ordinate of the other end is 9, find the abscissa of the other end? 6 or -10
f) ) The distance between A(1,3) and B(x,7) is 5. Calculate the possible value of x. 4or -2
2) a) Find the coordinates of the points on the x-axis which are at a distance of 5 units from the point (5,-4). (2,0),(8,0)
b) Find the coordinates of the points on the x-axis which are at a distance of 2√5 units from the point (7,-4). How many such points are there? (9,0),(5,0), 2
c) Find the coordinates of the points on the x-axis which are at a distances from the point (2,3) and (3/2, -1) are in the ratio 2:1. (0,0),(8/3,0)
d) ) Calculate the distance between A(7,3) and B on the x-axis whose abscissa is the 11. 5 units
3)a) Find the coordinates of the points on the y-axis which are at a distance of 13 units from the point (12,9). (0,4),(0,14)
b) Find the coordinates of the points on the y-axis which are at a distance of 10 units from the point (8,8). (0,2),(0,14)
4) a) A is a point on y-axis whose ordinate is 4 and B is a point on x-axis whose abscissa is -3. Find the length of the line segment AB. 5 units
b) Find the point/s which are at a distance of √10 units from the point (4,3) given that the ordinate of the point/s is twice the abscissa. (1,2) or (3,6)
c) A is a point on the y-axis whose ordinate is 5 and B is the point (-3,1), Calculate the length of AB. 5 units
d) What point(or points) in the y axis are at a distance of 10 units from the point (8,8). (0,2),(0,14)
5)a) The centre of a circle is (2a, a-7). Find the values of a if the circle passes through the point (11,-9) and has a diameter 10√2 units. 3,5
b) The centre of a circle is (2a-1, 3a+1) and passes through the point (-3,-1) and has a diameter 20 units, find the value/s of a. 2, -46/13
c) The centre of a circle is (x+2 , x-1). Find if the circle passes through (2,-2) and (8,-2). 3
d) Points A(-1,y) and B (5,7) lie on the circle with centre O(2, -3y). Find the value of y. Hence, find the radius of the circle. 7,-1, √793, 5
e) If the points A(4,3) and X(x,5) are on a circle with centre C(2,3), find the value of x. 2
EXERCISE -D
1) a) Show that the points A(6,6), B(2,3) and C(4,7) form a right angled triangle, Whose hypotenuse is AB.
b) A(6,0), B(-2,6) and C(12,8), Show that BC²= AB² + AC². Assign a special name to the triangle ABC.
c) Show that the points A(0,3), B(-2,1) and C(-1,4) form a right angled triangle.
d) Find a if the triangle formed by A(8,-10), B(7,-3) and C(0,a) is a right angled triangle at B. -4
e) Find a if the triangle formed by A(0,3), B(-2,a) and C(-1,4) is a right angled triangle at A. 1
f) The points (2,9),(a,5),(5,5) are the vertices of a triangle ABC right angled at B. Find the value/s of a and hence find the area of ∆ ABC. 2,5, 6sq. units
g) Show that the points (3,3),(9,0) and (12,21) are the vertices of a right angle triangle
2) a) Do the points P (3,2), Q(-2,-3) and (2,3) form a triangle? If so, name the type of triangle formed. triangle
b) Name the type of triangle formed by the points X(-5,6), Y(-4,-2) and C(7,5).
2)a) Show that the points A(2,-4), B(7,-1) and C(5,1) are vertices of an isosceles triangle. Name the equal sides. AB= AC
b) Check whether the points A(5,-2), B(6,4) and C(7,-2) are vertices of an isosceles triangle.
c) A(7,5), B(2,4) and C(6,10), show that AB= AC. Assign special name to the triangle ABC. Isosceles triangle
d) A(2,2), B(- 2,4) and C(2,6) are the vertices of a triangle ABC, Prove that ABC is an isosceles triangle.
e) If A(-3,2), B(x,y) and C(-1,4) are vertices of an isosceles triangle. Prove that x+ y= 1, given AB= BC
3) a) Show that the points X(1,1), Y(-1,-1) and Z(-√3,√3) form an equilateral triangle.
b) If two vertices of an equilateral triangle are (0,0),and (3,0), find the third vertex. (3/2, ±3√3/2)
4) a) Show that the points A(7,10), B(-2,5) and C(3,-4) are vertices of an isosceles right angled triangle.
b) Show that the points A(3,0), B(6,4) and C(-1,3) are vertices of an isosceles right angled triangle.
EXERCISE -E
1) Show that the point (4,4) is equidistance from the point A(1,0) and B(-1,4).
2) a) Find the value of K if the point P(2,4) is equidistant from the points A(5,K) and B(K,7). 3
b) If the point A(2,-4) is equidistant from the points P(3,8) and Q(-10,y), find the values of y. Also find distance PQ. -3, -5, √290, 13√2
c) If A(0,1) is equidistant from B(5,-3) and C(x,6), find the value of x. 4 or -4
d) If A(0,2) is equidistant from B(3,p) and C(p,5), find the value of p. 1
e) Find a relation between x and y such that the point (x,y) is equidistant from the points (7,1) and (3,5). x-y -2= 0
f) If the point P(x,y) is equidistant from the points A(a+ b, b -a) and B(A- b, a+ b), prove that bx = ay.
3)a) Find the point on the x-axis, which is equidistant from (2,-5) and (-2,9). (-7,0)
4)a) Find the coordinates of a point on the y-axis, which is equidistant from (4,0) and (4,12). (0,6)
b)Find the coordinates of a point on the y-axis, which is equidistant from A(6,5) and B(-4,3). (0,9)
5) Find the value of x such that PQ= QR where the coordinates P, Q and E are (6,-1),(1,3) and (x, 8). 5 or -3
6) The x-coordinate of a point X is twice its x-coordinate. If X is equidistant from the points Y(2,-5) and Z(-3,6), then find the coordinates of X. (16,8)
7) a) Find a relation between x and y such that the point (x,y) is equidistant from the points (3,6) and (-3,4). 3x+y -5= 0
8)a) By using distance formula, show that the points (4,2),(7,5), and (9,7) are collinear.
b) By using distance formula, show that the points (3,1),(6,4), and (8,6) are collinear.
EXERCISE - F
1) a) Show that the four points (2,-2),(8,4),(5,7) and (-1,1) are the vertices of a rectangle.
b) Show that the four points (0,-1),(-2,3),(6,7) and (8,3) are the vertices of a rectangle. Also find its area. 40
2)a) Show that the four points (0,5),(-2,-2),(5,0) and (7,7) are the vertices of a rhombus.
b) Show that the four points (3,0),(4,5),(-1,4) and (-2,-1) are the vertices of a rhombus. Also find the area of of the rhombus. 24 sq. units
c) Show that the four points (2,-1),(3,4),(-2,3) and (-3,-2) are the vertices of a rhombus. Also find its area. 24
3) Show that the four points (1,2),(-2,-1),(1,0) and (4,3) are the vertices of a parallelogram.
4) a) Show that the four points (2,1),(0,3),(-2,1) and (0,-1) are the vertices of a square.
b) Show that the four points (1,7),(4,2),(-1,-1) and (-4,4) are the vertices of a square.
c) Show that the four points (2,3),(-2,2),(-1,-2) and (3,-1) are the vertices of a square.
d) Show that the four points (2,1),(0,3),(-2,1) and (0,-1) are the vertices of a square. Also find its area. 8
5) A(-3,2), B(-5,-5), C(2,-3) and D(4,4) are the 4 points in the plane. Show that the ABCD is a rhombus but not a square.
6) Name the type of quadrilateral formed by the following points and give reasons for your answer:
i) (-1,-2),(1,0),(-1,2) and (-3,0). Square
ii) (4,5),(7,6),(4,3) and (1,2). ||gm
iii) (2,-2),(7,3),(11,-1) and (6,-6).
EXERCISE - G
1)a) Find the coordinates of the circumcentre of the triangle whose vertices are (4,6),(0,4),(6,2). (3,3)
b) Find the coordinates of the circumcentre of the triangle whose vertices are (8,6),(8,-2),(2,-2). Also, find its circumradius. (5,2), 5 units
c) Find the coordinates of the circumcentre of the triangle whose vertices are (5,1),(-3,-7),(7,-1). Hence find the circumradius. (2,-4), √34
d) Find the coordinates of the circumcentre of the triangle whose vertices are (2,-2),(8,-2),(8,6). (5,2)
e) Find the co-ordinates of the circumcentre of ∆ ABC with the vertices at A(3,0), B(-1,-6) and C(4,-1). Also, find its circum-radius. (1,-3), √13 units
2)a) Find the centre of a circle passing through the points (6,-6),(3,-7) and (3,3). Also find its radius. (3,-2),5 units
b) Find the centre of a circle passing through the points (5,1),(-3,-7) and (7,-1). (2,-4)
c) Find the coordinate of the centre of a circle which passes through the point A(0,0), B(-3,3) and C(5,-1). Also find the radius of the circle. (4,7), √65
3)a) The two opposite vertices of a square are (-1,2) and (3,2). Find the coordinates of the other two vertices. (1,0) and (1,4)
b) The ends of a diagonal of a square have coordinates (-2, p) and (p, 2). Find p if the area of the square is 40 sq. units. 6 or -6
Miscellaneous Exercise
1) KM is a straight line of 13 units. If K has the coordinates (2,5) and M has the coordinates (x,-7), find the possible values of x. 7 and -3
2) Find the point (or points) which are distance of √10 from the point (4,3), given that ordinate of the point (or points) is twice the abscissa . (1,2),(3,6)
3) Show that the point (7,3),(3,0),(0,-4) and (4,-1) are the vertices of a rhombus. Also find the area of the rhombus.
4) What point on the x-axis is equidistance from A(5,4) B(- 2,3)? (2,0)
5) What points on x-axis are at a distance of 17 units from the point (11,- 8)? (26,0) and (-4,0)
6) What points on y-axis are at a distance of 10 units from the point A(-8,4)? (0,10) and (0,-2)
7) A point P is at distance of √10 units from the point A (4,3). Find the coordinates of P, it being given that its ordinate is twice its abscissa . (3,6) and (1,2)
8) Show that the given points are collinear .
a) A(-2,3), B(1,2) and B(7,0).
b) A(3,-2), B(5,2) and C(8,8).
c) A(1,1),B(-2,7) and C(3,-3).
e) Show by distance formula that the points (-1,-1),(2,3) and (8,11) are collinear .
9) Show that the points A(7,10), B(- 2,5) and C(3,4) are the vertices of an isosceles right angled triangle. Also, find the area of the triangle. 53 sq. units
10) Show that the points P(1,1), Q(-1,1) and R(-√3, √3) are the vertices of an equilateral.
11) Show that the points L(2a, 4a), M(2a, 6a) and N(2a+ √3 a, 5a) are the vertices of an equilateral triangle with each side 2a.
12) Find a point equidistant from the point A(6,2), B(-1,3) and C(-3,-1). (2,-1)
13) Find the circumcentre of the triangle whose vertices are (-2,-3),(-1,0) and (7,-6). (3,-3)
14) Show that P(11,2) is the centre of the circle which passes through A(1,2), B(3,-4) and C(5,-6).
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