Frank solutions for Mathematics - Part 2 [English] Class 10 ICSE chapter 12 - Distance and Section Formulae [Latest edition]

Find the distance between the following pair of point in the coordinate plane :

(5 , -2) and (1 , 5)

Find the distance between the following pair of point in the coordinate plane.

(1 , 3) and (3 , 9)

Find the distance between the following pairs of point in the coordinate plane :

(7 , -7) and (2 , 5)

Find the distance between the following pairs of point in the coordinate plane :

(4 , 1) and (-4 , 5)

Find the distance between the following pairs of point in the coordinate plane :

(13 , 7) and (4 , -5)

Find the relation between x and y if the point M (x,y) is equidistant from R (0,9) and T (14 , 11).

Find the distance between P and Q if P lies on the y - axis and has an ordinate 5 while Q lies on the x - axis and has an abscissa 12 .

P and Q are two points lying on the x - axis and the y-axis respectively . Find the coordinates of P and Q if the difference between the abscissa of P and the ordinates of Q is 1 and PQ is 5 units.

Find the point on the x-axis equidistant from the points (5,4) and (-2,3).

A line segment of length 10 units has one end at A (-4 , 3). If the ordinate of te othyer end B is 9 , find the abscissa of this end.

Prove that the following set of point is collinear :

(5 , 5),(3 , 4),(-7 , -1)

Prove that the following set of point is collinear :

(5 , 1),(3 , 2),(1 , 3)

Prove that the following set of point is collinear :

(4, -5),(1 , 1),(-2 , 7)

Find the coordinate of O , the centre of a circle passing through A (8 , 12) , B (11 , 3), and C (0 , 14). Also , find its radius.

Find the coordinate of O , the centre of a circle passing through P (3 , 0), Q (2 , `sqrt 5`) and R (`-2 sqrt 2` , -1). Also find its radius.

Find the coordinates of O, the centre passing through A( -2, -3), B(-1, 0) and C(7, 6). Also, find its radius. 

The centre of a circle passing through P(8, 5) is (x+l , x-4). Find the coordinates of the centre if the diameter of the circle is 20 units.

Find the distance of the following point from the origin :

(5 , 12)

Find the distance of the following point from the origin :

(6 , 8)

Find the distance of the following point from the origin :

(8 , 15)

Find the distance of the following point from the origin :

(0 , 11)

Find the distance of the following point from the origin :

(13 , 0)

A(-2, -3), B(-1, 0) and C(7, -6) are the vertices of a triangle. Find the circumcentre and the circumradius of the triangle. 

P(5 , -8) , Q (2 , -9) and R(2 , 1) are the vertices of a triangle. Find tyhe circumcentre and the circumradius of the triangle.

x (1,2),Y (3, -4) and z (5,-6) are the vertices of a triangle . Find the circumcentre and the circumradius of the triangle.

Prove that the points (6 , -1) , (5 , 8) and (1 , 3) are the vertices of an isosceles triangle.

Prove that the points (1 ,1),(-4 , 4) and (4 , 6) are the certices of an isosceles triangle.

Prove taht the points (-2 , 1) , (-1 , 4) and (0 , 3) are the vertices of a right - angled triangle.

Prove that the points (7 , 10) , (-2 , 5) and (3 , -4) are vertices of an isosceles right angled triangle.

Prove that the points (1 , 1) , (-1 , -1) and (`- sqrt 3 , sqrt 3`) are the vertices of an equilateral triangle.

Prove that the points (0,3) , (4,3) and `(2, 3+2sqrt 3)` are the vertices of an equilateral triangle.

Prove that the points (5 , 3) , (1 , 2), (2 , -2) and (6 ,-1) are the vertices of a square.

Find the distance between the following point :

(p+q,p-q) and (p-q, p-q) 

Find the distance between the following point :

(sin θ , cos θ) and (cos θ , - sin θ)

Find the distance between the following point :

(sec θ , tan θ) and (- tan θ , sec θ)

Find the distance between the following point :

(Sin θ - cosec θ , cos θ - cot θ) and (cos θ - cosec θ , -sin θ - cot θ)

Prove that the points (4 , 6) , (- 1 , 5) , (- 2, 0) and (3 , 1) are the vertices of a rhombus.

Prove that the points (0 , 0) , (3 , 2) , (7 , 7) and (4 , 5) are the vertices of a parallelogram.

Prove that the points (0 , 2) , (1 , 1) , (4 , 4) and (3 , 5) are the vertices of a rectangle.

Prove that the points (a, b), (a + 3, b + 4), (a − 1, b + 7) and (a − 4, b + 3) are the vertices of a parallelogram. 

Prove that the points (0 , -4) , (6 , 2) , (3 , 5) and (-3 , -1) are the vertices of a rectangle.

ABCD is a square . If the coordinates of A and C are (5 , 4) and (-1 , 6) ; find the coordinates of B and D.

PQR  is an isosceles triangle . If two of its vertices are P (2 , 0) and Q (2 , 5) , find the coordinates of R if the length of each of the two equal sides is 3.

ABC is an equilateral triangle . If the coordinates of A and B are (1 , 1) and (- 1 , -1) , find the coordinates of C.

Find the distance of a point (7 , 5) from another point on the x - axis whose abscissa is -5.

Find the distance of a point (13 , -9) from another point on the line y = 0 whose abscissa is 1.

Find the distance of a point (12 , 5) from another point on the line x = 0 whose ordinate is 9.

Find the value of a if the distance between the points (5 , a) and (1 , 5) is 5 units .

Find the value of m if the distance between the points (m , -4) and (3 , 2) is 3`sqrt 5` units.

Find the relation between a and b if the point P(a ,b) is equidistant from A (6,-1) and B (5 , 8).

Find the coordinate of a point P which divides the line segment joining :

A (3, -3) and B (6, 9) in the ratio 1 :2. 

Find the coordinate of a point P which divides the line segment joining :

M( -4, -5) and N (3, 2) in the ratio 2 : 5. 

Find the coordinate of a point P which divides the line segment joining :

5(2, 6) and R(9, -8) in the ratio 3: 4. 

Find the coordinate of a point P which divides the line segment joining :

D(-7, 9) and E( 15, -2) in the ratio 4:7. 

Find the coordinate of a point P which divides the line segment joining :

A(-8, -5) and B (7, 10) in the ratio 2:3. 

In what ratio is the line joining (2, -4) and (-3, 6) divided by the line y = O ?

Find the ratio in which the line x = O divides the join of ( -4, 7) and (3, 0).
Also, find the coordinates of the point of intersection.

In what ratio does the point (1, a) divided the join of (−1, 4) and (4, −1) Also, find the value of a. 

(4, 2) and (-1, 5) are the adjacent vertices ofa parallelogram. (-3, 2) are the coordinates of the points of intersection of its diagonals. Find the coordinates of the other two vertices. 

Find the coordinates of point P which divides line segment joining A ( 3, -10) and B (3, 2) in such a way that PB: AB= 1.5. 

Find the ratio in which the line x = -2 divides the line segment joining (-6, -1) and (1, 6). Find the coordinates of the point of intersection. 

Find the ratio in which the line y = -1 divides the line segment joining (6, 5) and (-2, -11). Find the coordinates of the point of intersection. 

The line joining P (-5, 6) and Q (3, 2) intersects the y-axis at R. PM and QN are perpendiculars from P and Q on the x-axis. Find the ratio PR: RQ. 

B is a point on the line segment AC. The coordinates of A and B are (2, 5) and (1, 0). If AC= 3 AB, find the coordinates of C. 

Q is a point on the line segment AB. The coordinates of A and B are (2, 7) and (7, 12) along the line AB so that AQ = 4BQ. Find the coordinates of Q. 

The origin o (0, O), P (-6, 9) and Q (12, -3) are vertices of triangle OPQ. Point M divides OP in the ratio 1: 2 and point N divides OQ in the ratio 1: 2. Find the coordinates of points M and N. Also, show that 3MN = PQ. 

Find the points of trisection of the segment joining A ( -3, 7) and B (3, -2). 

A (2, 5), B (-1, 2) and C (5, 8) are the vertices of triangle ABC. Point P and Q lie on AB and AC respectively, such that AP: PB = AQ: QC = 1: 2. Calculate the coordinates of P and Q. Also, show that 3PQ = BC. 

A (30, 20) and B ( 6, -4) are two fixed points. Find the coordinates of a point Pin AB such that 2PB = AP. Also, find the coordinates of some other point Qin AB such that AB = 6 AQ. 

Show that the line segment joining the points (-3, 10) and (6, -5) is trisected by the coordinates axis.

Show that the lines x = O and y = O trisect the line segment formed by joining the points (-10, -4) and (5, 8). Find the points of trisection. 

Find the coordinates of the points of trisection of the line segment joining the points (3, -3) and ( 6, 9). 

Find the ratio in which the point P (2, 4) divides the line joining points (-3, 1) and (7, 6). 

Find the ratio in which the point R ( 1, 5) divides the line segment joining the points S (-2, -1) and T (5, 13). 

The points A, B and C divides the line segment MN in four equal parts. The coordinates of Mand N are (-1, 10) and (7, -2) respectively. Find the coordinates of A, B and C. 

Find the ratio in which the line segment joining A (2, -3) and B(S, 6) i~ divided by the x-axis. 

Find the ratio in which the line segment joining P ( 4, -6) and Q ( -3, 8) is divided by the line y = 0. 

In what ratio is the line joining (2, -1) and (-5, 6) divided by the y axis ?

Find the midpoint of the line segment joining the following pair of point :

(4,7) and (10,15) 

Find the midpoint of the line segment joining the following pair of point :

( -3, 5) and (9, -9) 

Find the midpoint of the line segment joining the following pair of point : 

(a+b, b-a) and (a-b, a+b) 

Find the midpoint of the line segment joining the following pair of point : 

(3a-2b, Sa+7b) and (a+4b, a-3b) 

Find the midpoint of the line segment joining the following pair of point : 

( a+3, 5b), (3a-1, 3b +4). 

A(6, -2), B(3, -2) and C(S, 6) are the three vertices of a parallelogram ABCD. Find the coordinates of the fourth vertex c. 

P( -2, 5), Q(3, 6 ), R( -4, 3) and S(-9, 2) are the vertices of a quadrilateral. Find the coordinates of the midpoints of the diagonals PR and QS. Give a special name to the quadrilateral. 

Three consecutive vertices of a parallelogram ABCD are A(S, 5), B(-7, -5) and C(-5, 5). Find the coordinates of the fourth vertex D. 

The points (2, -1), (-1, 4) and (-2, 2) are midpoints of the sides ofa triangle. Find its vertices.

If the midpoints of the sides ofa triangle are (-2, 3), (4, -3), (4, 5), find its vertices. 

If (-3, 2), (1, -2) and (5, 6) are the midpoints of the sides of a triangle, find the coordinates of the vertices of the triangle. 

Find the length of the median through the vertex A of triangle ABC whose vertices are A (7, -3), B(S, 3) and C(3, -1).

Find the centroid of a triangle whose vertices are (3, -5), (-7, 4) and ( 10, -2).

Two vertices of a triangle are (1, 4) and (3, 1). If the centroid of the triangle is the origin, find the third vertex. 

The mid-point of the line segment joining A (- 2 , 0) and B (x , y) is P (6 , 3). Find the coordinates of B.

A( 4, 2), B(-2, -6) and C(l, 1) are the vertices of triangle ABC. Find its centroid and the length of the median through C. 

A triangle is formed by line segments joining the points (5, 1 ), (3, 4) and (1, 1). Find the coordinates of the centroid.

The coordinates of the centroid I of triangle PQR are (2, 5). If Q = (-6, 5) and R = (7, 8). Calculate the coordinates of vertex P. 

Two vertices of a triangle are ( -1, 4) and (5, 2). If the centroid is (0, 3), find the third vertex. 

The midpoints of three sides of a triangle are (1, 2), (2, -3) and (3, 4). Find the centroid of the triangle. 

ABC is a triangle whose vertices are A(-4, 2), B(O, 2) and C(-2, -4). D. E and Fare the midpoint of the sides BC, CA and AB respectively. Prove that the centroid of the  Δ ABC coincides with the centroid of the Δ DEF.

Prove that the points A(-5, 4), B(-1, -2) and C(S, 2) are the vertices of an isosceles right-angled triangle. Find the coordinates of D so that ABCD is a square. 

The centre of a circle is (a+2, a-1). Find the value of a, given that the circle passes through the points (2, -2) and (8, -2).

Let A(-a, 0), B(0, a) and C(α , β) be the vertices of the L1 ABC and G be its centroid . Prove that 

GA² + GB² + GC² = `1/3` (AB² + BC² + CA²)

A(2, 5), B(-2, 4) and C(-2, 6) are the vertices of a triangle ABC. Prove that ABC is an isosceles triangle. 

A lies on the x - axis amd B lies on the y -axis . The midpoint of the line segment AB is (4 , -3). Find the coordinates of A and B .

P , Q and R are collinear points such that PQ = QR . IF the coordinates of P , Q and R are (-5 , x) , (y , 7) , (1 , -3) respectively, find the values of x and y.

A , B and C are collinear points such that AB = `1/2` AC . If the coordinates of A, B and C are (-4 , -4) , (-2 , b) anf (a , 2),Find the values of a and b.

The midpoint of the line segment joining the points P (2 , m) and Q (n , 4) is R (3 , 5) . Find the values of m and n.

The mid point of the line segment joining the points (p, 2) and (3, 6) is (2, q). Find the numerical values of a and b. 

The coordinates of the end points of the diameter of a circle are (3, 1) and (7, 11). Find the coordinates of the centre of the circle. 

AB is a diameter of a circle with centre 0. If the ooordinates of A and 0 are ( 1, 4) and (3, 6 ). Find the ooordinates of B and the length of the diameter.