Selina solutions for कन्साइस मैथमैटिक्स [अंग्रेजी] कक्षा १० आईसीएसई chapter 13 - Section Formula and Mid-Point Formula [Latest edition]

A point P divides the line segment joining the points A(1, 3) and B(5, 9) in the ratio 1: 2, the co-ordinates of the point P are ______.

AB is a line segment with A = (2, 4) and B = (6, 12). Point P lies on the line segment AB so that P = (3, x), then the ratio AP : PB is ______.

The ratio in which the join of (2, 4) and (10, 12) is divided by the line x = 7 is ______.

The line y = 4 divides the join of points (6, 7) and (4, −1) in the ratio ______.

The ratio in which the join of points (−2, 5) and (5, −2) is divided by y-axis is ______.

In what ratio does the point (1, a) divide the join of (–1, 4) and (4, –1)? Also, find the value of a.

In what ratio does the point (a, 6) divide the join of (–4, 3) and (2, 8)? Also, find the value of a.

In what ratio is the join of (4, 3) and (2, –6) divided by the x-axis? Also, find the co-ordinates of the point of intersection.

Find the ratio in which the join of (–4, 7) and (3, 0) is divided by the y-axis. Also, find the co-ordinates of the point of intersection.

Points A, B, C and D divide the line segment joining the point (5, –10) and the origin in five equal parts. Find the co-ordinates of B and D.

The line joining the points A (–3, –10) and B (–2, 6) is divided by the point P such that `(PB)/(AB) = 1/5`. Find the co-ordinates of P.

P is a point on the line joining A(4, 3) and B(–2, 6) such that 5AP = 2BP. Find the co-ordinates of P.

Calculate the ratio in which the line joining the points (–3, –1) and (5, 7) is divided by the line x = 2. Also, find the co-ordinates of the point of intersection.

Calculate the ratio in which the line joining A(6, 5) and B(4, –3) is divided by the line y = 2.

The point P (5, – 4) divides the line segment AB, as shown in the figure, in the ratio 2 : 5. Find the co-ordinates of points A and B. Given AP is smaller than BP.

Find the co-ordinates of the points of tri-section of the line joining the points (–3, 0) and (6, 6).

Show that the line segment joining the points (–5, 8) and (10, −4) is trisected by the co-ordinate axes.

Show that A (3, –2) is a point of trisection of the line segment joining the points (2, 1) and (5, −8). Also, find the co-ordinates of the other point of trisection.

The line segment joining the points M(5, 7) and N(–3, 2) is intersected by the y-axis at point L. Write down the abscissa of L. Hence, find the ratio in which L divides MN. Also, find the co-ordinates of L.

A (–3, 4), B (3, –1) and C (–2, 4) are the vertices of a triangle ABC. Find the length of line segment AP, where point P lies inside BC, such that BP : PC = 2 : 3.

The line segment joining A (2, 3) and B (6, –5) is intercepted by x-axis at the point K. Write down the ordinate of the point K. Hence, find the ratio in which K divides AB. Also, find the coordinates of the point K.

The line segment joining A(4, 7) and B(−6, −2) is intercepted by the y – axis at the point K. write down the abscissa of the point K. hence, find the ratio in which K divides AB. Also, find the co-ordinates of the point K.

The line joining P(–4, 5) and Q(3, 2) intersects the y-axis at point R. PM and QN are perpendicular from P and Q on the x-axis Find:

1. the ratio PR : RQ
2. the coordinates of R.
3. the area of the quadrilateral PMNQ.

In the given figure, line APB meets the x-axis at point A and y-axis at point B. P is the point (−4, 2) and AP : PB = 1 : 2. Find the co-ordinates of A and B.

Given a line segment AB joining the points A(−4, 6) and B(8, −3). Find:

1. the ratio in which AB is divided by the y-axis.
2. find the coordinates of the point of intersection.
3. the length of AB.

If P(–b, 9a – 2) divides the line segment joining the points A(–3, 3a + 1) and B(5, 8a) in the ratio 3: 1, find the values of a and b.

Point A (3, 4) is the centre of a circle. If one of its diameters has one end as (7, 8); the other end of this diameter is ______.

Point A lies on x-axis and point B lies on y-axis. If P(2, −2) bisects the line segment AB, the co-ordinates of A are ______.

In parallelogram ABCD, A = (6, 0), B = (12, −4) and C = (4, −4); then the cоordinates of vertex D are ______.

The point P (2, −7) is reflected in the point (0, 3); the co-ordinates of the image of point P are ______.

The co-ordinates of the centroid of a triangle with vertices (−6, −3), (0, 0) and (12, −6) are ______.

Points A and B have co-ordinates (3, 5) and (x, y) respectively. The mid-point of AB is (2, 3). Find the values of x and y.

Given M is the mid-point of AB, find the co-ordinates of A; if M = (1, 7) and B = (–5, 10).

Given M is the mid-point of AB, find the co-ordinates of B; if A = (3, –1) and M = (–1, 3).

P(–3, 2) is the mid-point of line segment AB as shown in the given figure. Find the co-ordinates of points A and B.

(–5, 2), (3, −6) and (7, 4) are the vertices of a triangle. Find the length of its median through the vertex (3, −6).

Given a line ABCD in which AB = BC = CD, B = (0, 3) and C = (1, 8). Find the co-ordinates of A and D.

A(2, 5), B(1, 0), C(−4, 3) and D(–3, 8) are the vertices of quadrilateral ABCD. Find the co-ordinates of the mid-points of AC and BD. Give a special name to the quadrilateral.

P(4, 2) and Q(–1, 5) are the vertices of parallelogram PQRS and (–3, 2) are the co-ordinates of the point of intersection of its diagonals. Find co-ordinates of R and S.

The points (2, –1), (–1, 4) and (–2, 2) are mid-points of the sides of a triangle. Find its vertices.

Points A(–5, x), B(y, 7) and C(1, –3) are collinear (i.e. lie on the same straight line) such that AB = BC. Calculate the values of x and y.

Points P(a, −4), Q(−2, b) and R(0, 2) are collinear. If Q lies between P and R, such that PR = 2QR, calculate the values of a and b.

The co-ordinates of the centroid of a triangle PQR are (2, –5). If Q = (–6, 5) and R = (11, 8); calculate the co-ordinates of vertex P.

A(5, x), B(−4, 3) and C(y, –2) are the vertices of the triangle ABC whose centroid is the origin. Calculate the values of x and y.

If P(4, 3) is mid-point of line segment AB, then ______.

Points A and B have co-ordinates (4, y) and (x, 8). The mid-point of AB = (0,0); the values of x and y are ______.

Two vertices of a triangle are (−5, 3) and (4, −7). If centroid of the triangle is (−1, 2) then the third vertex is ______.

Points A, B, C and D divide the join of O(0, 0) and P(0, 10) into five equal parts. The co-ordinates of point D are ______.

Line segment joining points (4, 3) and (1, −2) is divided by the point (y, 0) in the ratio ______.

The point P divides the line segment joining the points (1, 2) and (−1, 2) internally in the ratio 1 : 2. 

Assertion (A): The co-ordinates of point P = (1, 6). 

Reason (R): If point P divides line segment joining the points \[(x_1, y_1)\] and \[(x_2, y_2)\] in the ratio \[m_1:m_2\] then \[\mathrm{P}=\left(\frac{m_1 x_2+m_2 x_1}{m_1+m_2}, \frac{m_1 y_2+m_2 y_1}{m_1+m_2}\right)\]

Point P(x, y) is equidistant from points A(7, 1) and B(3, 5).

Statement (1): \[x=\frac{7+3}{2}\] and \[y=\frac{1+5}{2}\]. 

Statement (2): \[(x-7)^{2}+(y-1)^{2}=(x-3)^{2}+(y-5)^{2}\]

Given a triangle ABC in which A = (4, −4), B = (0, 5) and C = (5, 10). A point P lies on BC such that BP : PC = 3 : 2. Find the length of line segment AP.

A(20, 0) and B(10, –20) are two fixed points. Find the co-ordinates of the point P in AB such that : 3PB = AB. Also, find the co-ordinates of some other point Q in AB such that : AB = 6 AQ.

A(–8, 0), B(0, 16) and C(0, 0) are the vertices of a triangle ABC. Point P lies on AB and Q lies on AC such that AP : PB = 3 : 5 and AQ : QC = 3 : 5. Show that : PQ = `3/8` BC.

A line segment joining A`(-1,5/3)` and B(a, 5) is divided in the ratio 1 : 3 at P, the point where the line segment AB intersects the y-axis.

1. Calculate the value of ‘a’.
2. Calculate the co-ordinates of ‘P’.

In what ratio is the line joining A(0, 3) and B(4, –1) divided by the x-axis? Write the co-ordinates of the point where AB intersects the x-axis.

The mid-point of the segment AB, as shown in diagram, is C(4, –3). Write down the co-ordinates of A and B.

AB is a diameter of a circle with centre C = (–2, 5). If A = (3, –7), find

1. the length of radius AC.
2. the coordinates of B.

The mid point of the line segment joining (4a, 2b – 3) and (−4, 3b) is (2, –2a). Find the values of a and b.

1. Write down the co-ordinates of the point P that divides the line joining A(−4, 1) and B(17, 10) in the ratio 1 : 2.
2. Calculate the distance OP, where O is the origin.
3. In what ratio does the y-axis divide the line AB?

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

M is the mid-point of the line segment joining the points A(–3, 7) and B(9, –1). Find the coordinates of point M. Further, if R(2, 2) divides the line segment joining M and the origin in the ratio p : q, find the ratio p : q.

Calculate the ratio in which the line joining A(−4, 2) and B(3, 6) is divided by point P(x, 3). Also, find

1. x
2. length of AP.

Find the ratio in which the line 2x + y = 4 divides the line segment joining the point P(2, –2) and Q(3, 7).

If the abscissa of a point P is 2, find the ratio in which this point divides the line segment joining the point (−4, 3) and (6, 3). Also, find the co-ordinates of point P.

Find the image of the point A(5, –3) under reflection in the point P(–1, 3).

M is the mid-point of the line segment joining the points A(0, 4) and B(6, 0). M also divides the line segment OP in the ratio 1 : 3. Find : 

1. co-ordinates of M
2. co-ordinates of P
3. length of BP

A(3, 1), B(y, 4) and C(1, x) are vertices of triangle ABC and G(3, 4) is its centroid. Find the values of x and y. Also, find the length of side BC.

Three friends Govind, Rishi and Kanika are participating in a Treasure Hunt organized in their school playground. The playground is mapped using a coordinate grid where each square represents 1 meter.

At a certain point in the game, they each stand at different spots waiting for their next clue. Their positions are recorded on the grid as points:

• Govind is at point P
• Rishi is at point Q
• Kanika is at point R 

The coordinate map is shown alongside. 

Based on the above information, answer the following questions:

1. Is Q the midpoint of segment PR? Justify your answer.
2. A new clue directs them to reach point M, which divides segment PQ in the ratio 2 : 3. Find the coordinates of M.