**Section And Mid Point Formula** Concise Maths Solutions Chapter-13.Solutions of Exercise – 13 (A), Exercise – 13 (B),Exercise – 13 (C), Exercise – 13 (D) for Concise Selina Maths of ICSE Board Class 10th. Concise Solutions **Section And Mid Point Formula** Chapter – 13 for ICSE Maths Class 10 is available here. All Solutions of Concise Selina of Chapter-13 **Section And Mid Point Formula** has been solved according instruction given by council. This is the Solutions of Chapter-13 **Section And Mid Point Formula** for ICSE Class 10th. ICSE Maths text book of Concise is In series of famous ICSE writer in maths publications. Concise is most famous among students.

**Section And Mid Point Formula Concise Maths Solutions Chapter-13**

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**How to Solve Concise Maths Selina Publications Chapter-13 Section And Mid Point Formula**

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**EXERCISE – 13( A)** **Solutions of Concise Maths Section And Mid Point Formula**

** **

**Question 1.**

**Calculate the co-ordinates-of the point P which divides the line segment joining:**

**(i) A (1, 3) and B (5, 9) in the ratio 1 : 2**

**(ii) A (-4, 6) and B (3, -5) in the ratio 3 : 2**

**Answer 1**

(i) Let co-ordinates of P be (x,y)

**Question 2.**

**In what ratio is the line joining (2, -3) and (5, 6) divided by the x-axis ?**

**Answer 2**

Let the point P (x, 0) divides in the ratio of m_{1} : m_{2} line joining the points A (2, -3) and B (5, 6)

**Question 3.**

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

**Answer 3**

Let the point P (0, y) divides the line joining the points A (2, -4) and (-3, 6) in the ratio of m_{1} : m_{2}

**Question 4.**

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

**Answer 4**

Let the point P (1, a) divides the line joining the points (-1, 4) and (4, -1) in the ratio of m_{1} : m_{2}

**Question 5.**

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

**Answer 5**

Let the point P (a, 6) divides the line joining the points A (-4, 3), B (2, 8) in the ratio of m_{1} : m_{2}

**Question 6.**

**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.**

**Answer 6**

**QUESTION 7**

**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.**

**Answer 7**

Let, the points (0, y) be the point of intersection which divides the line joining the points A (-4, 7) and B (3, 0)

**Question 8.**

**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 A, B, C and D.**

**Answer 8**

Points A, B, C and D divide the line segment joining the points (5, -10) and origin (0, 0) in five equal parts

Let co-ordinates of A be (x, y) which divides PO in the ratio of 1 : 4

**Question 9.**

**The line joining the points A (-3, -10) and B (-2, 6) is divided by the point P such that = = , find the co-ordinates of P.**

**Answer 9 **

Let the co-ordinates of P be (x, y) which divides the line joining the points A (-3,-10) and B (-2,6) in the ratio of AP : PB i.e. (5 – 1) : 1 or 4 : 1

**Question 10.**

**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.**

**Answer 10**

**Question 11.**

**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.**

**Answer 11**

Let the point P (2, y) divides the line joining the points A (-3, -1) and B (5, 7) in the ratio of m_{1} : m_{2}

**Question 12.**

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

**Answer 12**

Let the point P (x, 2) divides the line joining the points A (6, 5) and B (4, -3) in the ratio of m_{1} : m_{2}

**Question 13.**

**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.**

**Answer 13**

From the figure, the line AB intersects x-axis at A and y-axis at B.

Let the co-ordinates of A (x, 0) and B (0, y) and P (5, -4) divides it in the ratio of 2 : 5

**Question 14.**

**Find the co-ordinates of the points of trisection of the line joining the points (-3, 0) and (6, 6).**

**Answer 14**

**Question 15.**

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

**Answer 15**

Let the points A (-5, 8) and B (10, -4).

Let P and Q be the two points on the axis which trisect the line joining the points A and B.

AP = PQ = QB

AP : PB = 1 : 2 and AQ : QB = 2 : 1

**Question 16.**

**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.**

**Answer 16**

Let A and B are the points of trisection of the line segment joining the points P (2, 1) and Q (5, -8), then

PA = AB = BQ.

PA : AQ = 1 : 2 and PB : BQ = 2 : 1

**Question 17.**

**If A = (-4, 3) and B = (8, -6)**

**(i) find the length of AB**

**(ii) In what ratio is the line joining A and B, divided by the x-axis ?**

**Answer 17**

**Question 18.**

**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.**

**Answer 18**

**Question 19.**

**A (2, 5), B (-1, 2) and C (5, 8) are the co-ordinates of the vertices of the triangle ABC. Points P and Q lie on AB and AC respectively, such that:**

**AP : PB = AQ : QC = 1 : 2.**

**(i) Calculate the co-ordinates of P and Q.**

**(ii) Show that PQ = BC.**

**Answer 19**

**Question 20.**

**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.**

**Answer 20**

**Question 21.**

**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 co-ordinates of the point K. [1990, 2006]**

**Answer 21 **

Let the line segment Intersect the x-axis at the point P

Co-ordinates of P are (x, 0)

Let P divide the line segment in the ratio K : 1 then

**Question 22.**

**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.**

**Solution:**

**Answer 22**

Points A (4, -7), B (-6, -2) are joined which intersects y-axis at K. abscissa of K will be 0

Let the coordinates of K be (0, y) and K divides AB line segment in the ratio m_{1} : m_{2}

**Question 23.**

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

**(i) The ratio PR: RQ.**

**(ii) The co-ordinates of R.**

**(iii) The areas of the quadrilateral PMNQ. [2004]**

**Answer 23**

(i) Let divides the line joining the points P (-4, 5) and Q (3, 2) in the ratio k : 1

**Question 24.**

**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.**

**Answer 24**

Let the co-ordinates of A be (x1, 0) (as it lies on x-axis)

and co-ordinates of B be (0, y2)

and co-ordinates of P are (-4, 2)

AP : PB = 1 : 2 i.e. m1 = 1, m2 = 2

Now, P divides AB in the ratio m_{1} : m_{2} or 1 : 2

**Question 25.**

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

**(i) the ratio in which AB is divided by the y-axis.**

**(ii) find the coordinates of the point of intersection.**

**(iii) the length of AB.**

**Answer 25**

(i) Let the y-axis divide AB in the ratio m : 1

#### Question 26

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.

**Answer 26**

Take (x_{1} , y_{1}) = (-3, 3a + 1) ; (x_{2} , y_{2}) = B(5, 8a) and

(x, y) = (-b, 9a – 2)

Here m_{1} = 3 and m_{2} =1

**Solutions of Selina Publishers Concise Maths Section And Mid Point Formula,**** EXERCISE- 13 B**

** **

**Question 1.**

**Find the mid-point of the line segment joining the points:**

**(i) (-6, 7) and (3, 5)**

**(ii) (5, -3), (-1, 7)**

**Answer 1**

Let P (x, y) be the mid-point in each case

**Question 2**

**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.**

**Answer 2**

Co-ordinates of A (3, 5), B (x, y) and mid-point M (2, 3)

**Question 3**

**A (5, 3), B (-1, 1) and C (7, -3) are the vertices of ABC. If L is the mid-point of AB and M is the mid-point of AC, show that LM = BC.**

**Answer 3**

**Question 4**

**Given M is the mid-point of AB, find the co-ordinates of:**

**(i) A; if M = (1, 7) and B = (-5, 10),**

**(ii) B; if A = (3, -1) and M (-1, 3).**

**Answer 4**

M is the mid-point of AB.

(i) Let A = (x, y), M = (1, 7) and B = (-5, 10)

**Question 5**

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

**Answer 5**

Point A is on y-axis

its abscissa is zero and point B is on x-axis

its ordinate is zero.

Now, let co-ordinates of A are (0, y) and ofB are (x, 0) and P (-3, 2) is the mid-point

**Question 6**

**.****In the given figure, P (4, 2) is the mid point of line segment AB. Find the co-ordinates of A and B.**

**Answer 6**

Points A and B are on x-axis and y-axis respectively

Ordinate of A is zero and abscissa of B is zero.

Let co-ordinates of A be (x, 0) and B (0, y)

and P (4, 2) is the mid-point

**Question 7**

**(-5, 2), (3, -6) and (7, 4) arc the vertices of a triangle. Find the length of its median through the vertex (3, -6) and (7, 4).**

**Answer 7**

Let A (-5, 2), B (3, -6) and C (7, 4) are the vertices of a ABC

Let L,M and N are the mid-points of sides BC, CA and AB respectively of ABC.

L is the mid-point of BC.

Co-ordinates of L will be

**Question 8**

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

**Answer 8**

**Question 9**

**One end of the diameter of a circle is (-2, 5). Find the co-ordinates of the other end of it, if the centre of the circle is (2, -1).**

**Answer 9**

**Question 10**

**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.**

**Answer 10**

Co-ordinates of A = (2, 5), B = (1, 0), C = (-4, 3) and D = ( 3, 8)

Let the mid-point of AC is P (x1, y1) Co-ordinates of mid-point of AC will be

Co-ordinates of mid-points AC and BD are the same..

The quadrilateral is a parallelogram.

**Question 11**

**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 the co-ordinates of R and S.**

**Answer 11**

In the parallelogram PQRS and qo-ordinates of P are (4, 2) and of Q are (-1, 5).

The diagonals of || gm AC and BD intersect each other at O (-3, 2)

**Question 12**

**A (-1, 0), B (1, 3) and D (3, 5) are the vertices of a parallelogram ABCD. Find the co-ordinates of vertex C.**

**Answer 12**

Vertices of a parallelogram ABCD are A (-1, 0), B (1, 3) and D(3, 5)

Let co-ordinates of C be (x, y)

Let the diagonals AC and BD bisect each other at O. Then O is the mid-point of AC as well as of BD.

Co-ordinates of O, the mid-point of BD will be

**Question 13**

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

**Answer 13**

Let D, E and F are the mid-points of sides BC, CA and AB of a ABC respectively.

Co-ordinates of A are (-5, 7), of B are (1, -3) and of C are (3, 1)

**Question 14**

**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.**

**Answer 14**

**Question 15**

**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’:**

**Answer 15**

**Question 16**

**Calculate the co-ordinates of the centroid of the triangle ABC, if A = (7, -2), B = (0, 1) and C = (-1, 4).**

**Answer 16**

**Question 17**

**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.**

**Answer 17**

**Question 18**

**.****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.**

**Answer 18**

** EXERCISE – 13 (C) Section And Mid Point Formula Concise Maths Solutions Selina Publishers**

** **

**Question 1**

**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.**

**Answer 1**

B (0, 5), C (5, 10) and BP : PC = 3 : 2 Co-ordinates of P will be

**Question 2**

**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 = 6AQ.**

**Answer 2**

(i) A (20, 0), B (10, – 20)

**Question 3**

**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 = BC.**

**Answer 3**

**Question 4**

**Find the co-ordinates of points of trisection of the line segment joining the points (6, -9) and the origin.**

**Answer 4**

Points are A (6, -9) and O (0,0) let P and Q are points, which trisect AO

**Question 5**

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

**(i) Calculate the value of ‘a’.**

**(ii) Calculate the co-ordinates of ‘P’. (1994)**

**Answer 5**

**Question 6**

**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. [1993]**

**Answer 6**

Let the ratio be m_{1} : m_{2} when the x-axis intersects the line AB at P.

**Question 7**

**The mid point of the segment AB, as shown in diagram, is C (4, -3). Write down the co-ordinates of A and B. (1996)**

**Answer 7**

Let co-ordinates of A (x, 0) and B (0, y) and C (4, -3) the mid point of AB.

**Question 8**

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

**(i) the length of radius AC**

**(ii) the coordinates of B.**

**Answer 8**

**Question 9**

**Find the co-ordinates of the centroid of a triangle ABC whose vertices are A (- 1, 3), B (1, – 1) and C (5, 1)**

**Answer 9**

**Question 10**

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

**Answer 10**

Let A and B are two points and P is its mid point then A is (4a, 2b -3), B(-4, 2b) and P is (2, -2a)

**Question 11**

**The mid point of the line segment joining (2a, 4) and (-2, 2b) is (1, 2a + 1). Find the value of a and b.**

**Answer 11**

The midpoint of the line segment joining (2a, 4) and (-2, 2b) is (1, 2a + 1)

**Question 12**

**(i) 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.**

**(ii) Calculate the distance OP, where O is the origin.**

**(iii) In what ratio does the y-axis divide the line AB ? [ICSE 1995]**

**Answer 12**

Point P, divides a line segment giving the points A (-4, 1) and B (17, 10) is the ratio 1 : 2.

**Question 13**

**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. [1992]**

**Answer 13**

In ABC, the co-ordinates of A, B and C are (-5, 4), B(-1, -2) and C (5, 2) respectively.

ABC is also a right-angled triangle.

Hence ABC is an isosceles right angled triangle,

Let D be the fourth vertex of square ABCD and co-ordinates of D be (x,y)

Since the diagonals of a square bisect each other and let O be the point of intersection of AC and BD.

O is mid-point of AC as well as BD.

**Question 14**

**M is the mid-point of the line segment joining the points A (-3, 7) and B (9, -1). Find the co-ordinates 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.**

**Answer 14**

Two points are given A (-3, 7) and B (9, -1)

M is the mid-point of line joining AB.

Co-ordinates of M wll be

**Question 15**

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

**(i) x**

**(ii) Length of AP. (2014)**

**Answer 15**

Let ratio = k : 1

**Question 16**

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

**Answer 16**

Let the given line 2x + y = 4 divides the line segment joining the points P (2, -2) and Q (3,7) in the ratio k : 1 at a point (x, y) on it.

**Question 17**

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

**Answer 17**

Abscissa of a point P is 2

Let co-ordinates of point P be (2, y)

Let point P (2, y) divides the line segment joining the points (-4, 3) and (6, 3) in the ratio k : 1

**Question 18**

**The line joining the points (2, 1) and (5, -8) is trisected at the points P and Q. If point P lies on the line 2x – y + k = 0, find the value of k, Also, find the co-ordinates of point Q.**

**Answer 18**

A line joining the points (2, 1) and (5, -8) is trisector at P and Q.

**Question 19**

**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:**

**(i) co-ordinates of M**

**(ii) co-ordinates of P**

**(iii) length of BP**

**Answer 19**

M is mid point of the line segment joining the points A (0, 4) and B (6, 0)

M divides the line segment OP in the ratio 1 : 3

**Question 20**

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

**Answer 20**

Image of the point A (5, -3) under reflection in the point P (-1, 3)

Let B (x, y) be the point of reflection of A (5, -3) under P(-1, 3)

**Question 21**

**A (-4, 2), B (0, 2) and C (-2, -4) are vertices of a triangle ABC. P, Q and R are mid-points of sides BC, CA and AB respectively. Show that the centroid of PQR is the same as the centroid of ABC.**

**Answer 21**

A (-4, 2), B (0, 2) and C (-2, -4) are the vertices of ABC.

P, Q and R are the mid-points of the sides BC, CA and AB respectively.

G is the centroid of medians AP, BQ and CR.

Co-ordinates of G are

A(3, 1), B(y, 4) and C(1, x) are vertices of a triangle ABC. P, Q and R are mid – points of sides BC, CA and AB respectively. Show that the centroid of ΔPQR is the same as the centroid ΔABC.

**Answer **

P, Q and R are the mid points of the sides BC, CA and AB.

By mid – point formula, we get

From (i) and (ii), we get

#### End of Chapter-13 Section And Mid Point Formula for Concise Maths Solutions

** **

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