**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.Solution:**

(i) Let us assumed that, co-ordinates of the point P be (x, y)

We know that

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

Solution:

It is given that, the joining points A (2,-3) and B (5,6) divided by the x-axis by the point P(x,0).

Let the ratio is k:1.

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

Solution:

It is given that, the joining points A (2,-4) and B (-3, 6) divided by the x-axis by the point P(0, x).

Let the ratio is k:1.

**Question **4. In what ratio does the point (1, a) divided the join of (-1, 4) and (4, -1)? Also, find the value of a.**Solution:**

It is given that, the joining points A (-1, 4) and B (4, -1) divided by the x-axis by the point P (1, a). Let the ratio is k: 1

**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.

Solution:

It is given that, the joining points A (-4, 3) and B (2, 8) divided by the x-axis by the point P (a, 6).

Let the ratio is k: 1.

** 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. Solution:**It is given that, the joining points A (4, 3) and B (2, -6) divided by the x-axis by the point P (x, 0).

Let the ratio is k: 1.

Hence, the required ratio is 1: 2 and the value of a is 10/3.

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

Solution:

It is given that, the joining points A (-4, 7) and B (3, 0) divided by the x-axis by the point P (0, y).

Let the ratio is k: 1

**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.**Solution:**

The coordinates of point A is (3,-6).

Point C divides the line PO in the ratio 3:2.

**Question **9. The line joining the points A (-3, -10) and B (-2, 6) is divided by the point P such that

**Solution:**

Let us assumed that, the co-ordinates of Point P (x, y)

**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.

Solution:

It is given that, 5AP = 2BP

AP/BP=2/5

Co-ordinates of the point P are

**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.

Solution:

It is given that, the points are (-3, -1) and (5, 7) is divided by the line x = 2, will be of the type (2, y).

**Question **12. Calculate the ratio in which the line joining A (6, 5) and B (4, -3) is divided by the line y = 2.

Solution:

It is given that, the point A (6, 5) and (4, -3) is divided by the line y = 2, will be of the type (x, 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.

**Solution:**It is given that, the point P (5, -4) divides the line segment AB, as shown in the figure, in the ratio 2: 5.

Here, Point A lies on x-axis, so the coordinates of A be (x, 0) and B lies on y-axis so, the coordinates of B be (0, y).

We know that, the section formula,

**Question **14. Find the co-ordinates of the points of trisection of the line joining the points (-3, 0) and (6, 6).

Solution:

It is given that, the co-ordinates of the points of trisection of the line joining the points (-3, 0) and (6, 6). AP=PQ=QB

Here,

AP:PB =1:2

Coordinates of the points P,

**Question **15. Show that the line segment joining the points (-5, 8) and (10, -4) is trisected by the co- ordinate axes.

Solution:

It is given that, the co-ordinates of the points of trisection of the line joining the points (-5, 8) and (10, -4). AP=PQ=QB

Here,

AP:PB =1:2

Coordinates of the points P,

**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.**Solution:**

It is given that, the co-ordinates of the points of trisection of the line joining the points (2, 1) and (5, -8). PA=AB=BQ

Here,

PA:AQ=1:2

Coordinates of the point A.

**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?Solution:**

(i) It is given that,

Points of A = (-4, 3) and B = (8, -6)

**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.

Solution:

It is given that L lines on y-axis coordinate on y axis are 0. Let P be the point on the x-axis that divides AB in the ratio k: 1

**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 = 1/3 BC.Solution:**

(i) It is given that, A (2, 5), B (-1, 2) and C (5, 8) are the co-ordinates of the vertices of the triangle ABC.

Coordinates of P are,

**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.

Solution:

It is given that, the ratio of BP:PC is 2:3.

Here,

Coordinates of P are,

**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.

Solution:

It is given that, the line segment joining A (2, 3) and B (6, -5) is intercepted by x-axis at the point K, So, its ordinate is 0.

Let us assumed that, point k(x,0) divides AB in the ratio k:1

**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:

It is given that, the line segment joining A (4, 7) and B (-6, -2) is intercepted by the y-axis at the point K.

Let us assumed that, point k(0,y) divides AB in the ratio k:1.

**Question **23. The line joining P (-4, 5) and Q (3, 2) intersects the y-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 area of the quadrilateral PMNQ.** **Solution:**

(i) It is given that, the line joining P (-4, 5) and Q (3, 2) intersects the y-axis at point R. Let us assumed that, point R (0,y) PQ 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.

**Solution:**

It is given that, line APB meets the x-axis at point A and y-axis at point B. Let the co-ordinates of A and B be (x, 0) and (0, y) respectively. P is the point (-4, 2) and AP: PB = 1: 2.

By Section formula,

**Question **25. 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.

Solution:

It is given that, (x1, y1) = (-3, 3a+1); (x2, y2) = B (5, 8a) and (x, y) = (-b, 9a – 2)

m_1=3 and m_2=1

Coordinate of P(x, y)

### Exercise 13B

**Question **1. Find the mid-point of the line segment joining the points:**(i) (-6, 7) and (3, 5)****(ii) (5, -3) and (-1, 7)** **Solution:**

(i) It is given that, A(-6, 7) and B(3, 5)

**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.

Solution:

It is given that, points A and B have co-ordinates (3, 5) and (x, y) and mid-point of AB is (2, 3).

**Question **3. A (5, 3), B (-1, 1) and C (7, -3) are the vertices of triangle ABC. If L is the mid-point of AB and M is the mid-point of AC, show that LM = 1/2 BC.

Solution:

It is given that, A (5, 3), B (-1, 1) and C (7, -3) are the vertices of triangle ABC. L is the mid-point of AB and M is the mid-point of AC.

Here,

Co-ordinates of L are,

**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). Solution:**

(i) It is given that, M is the mid-point of AB.

Let the co-ordinates of A be (x, y)

**Question **5. 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.

**Solution:**It is given that, P (-3, 2) is the mid-point of line segment AB. Let us assumed that the co-ordinated of point A be (0,y) and B be (x,0).

**Question **6. In the given figure, P (4, 2) is mid-point of line segment AB. Find the co-ordinates of A and B.

**Solution:**It is given that, P (4, 2) is the mid-point of line segment AB. Let us assumed that the co-ordinated of point A be (x,0) and B be (0,y).

**Question **7. (-5, 2), (3, -6) and (7, 4) are the vertices of a triangle. Find the lengths of its median through the vertex (3, -6).**Solution:**

It is given that, A (-5, 2), B (3, -6) and C (7, 4) be the vertices of the given triangle.

Let us assumed that, D is the median of AB, E is the median of BC and F is the median of CA.

The median of the triangle through the vertex B (3,-6) is BE. Here, coordinates of B (3, -6) and coordinate of E (1,3) We know that the distance formula,

**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.**Solution:**

**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).

Solution:

It is given that, the coordinated of the diameter of a circle is (-2, 5) and the centre of the circle is (2,-1).

Let the required co-ordinates of the other end of mid-point be (x, y).

**Question **10. A (2, 5), B (1, 0), C (-4, 3) and D (-3, 8) are the vertices of a quadrilateral ABCD. Find the co-ordinates of the mid-points of AC and BD. Give a special name to the quadrilateral.

Solution:

It is given that, A (2, 5), B (1, 0), C (-4, 3) and D (-3, 8) are the vertices of a quadrilateral ABCD.

Here, Co-ordinates of the mid-point of AC,

**Question **11. P (4, 2) and Q (-1, 5) are the vertices of a parallelogram PQRS and (-3, 2) are the co- ordinates of the points of intersection of its diagonals. Find the coordinates of R and S.

Solution:

It is given that, the coordinates of P(4,2) and Q(-1,5) and (-3, 2) are the co-ordinates of the points of intersection of its diagonals. Let us assumed that, the co-ordinates of P(4,2) and R(x, y)

**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.

Solution:

Let the coordinates of vertex C be (x,y).

ABCD is a parallelogram.

Mid-point of AC = Mid-point of BD

**Question **13. The points (2, -1), (-1, 4) and (-2, 2) are mid-points of the sides of a triangle. Find its vertices.**Solution:**

It is given that, the points (2, -1), (-1, 4) and (-2, 2) are mid-points of the sides of a triangle.

Let us assumed that, A(x_{1},y_{1}), B(x_{2},y_{2}), C(x_{3},y_{3}) are the co-ordinates of the vertices of ∆ABC.

By adding equation (1), (3) and (5), we get,

𝑥_{1} + 𝑥_{2 }+ 𝑥_{1} + 𝑥_{3} + 𝑥_{2} + 𝑥_{3 }= 4 + (−2) + (−6)

2𝑥_{1} + 2𝑥_{2} + 2𝑥_{3} = 4 − 2 − 6

2(𝑥_{1} + 𝑥_{2} + 𝑥_{3}) = 4 − 2 − 6

2(𝑥_{1} + 𝑥_{2} + 𝑥_{3}) = −2

𝑥_{1} + 𝑥_{2} + 𝑥_{3} = −1

Put the value of 𝑥_{1} + 𝑥_{2 }in above equation,

4 + 𝑥_{3} = −1

𝑥_{3} = −1 − 4

𝑥_{3 }= −5

Put the value of 𝑥_{3} in equation (3)

−2 = 𝑥1 + (−5)

−2 = 𝑥_{1} − 5

−2 + 5 = 𝑥1

3 = 𝑥_{1}

𝑥_{1} = 3

Put the value of 𝑥_{3 }in equation (5)

−4 = 𝑥_{2} + 𝑥_{3}

−4 = 𝑥_{2} + (−5)

−4 = 𝑥_{2} − 5

−4 + 5 = 𝑥_{2}

1 = 𝑥_{2}

𝑥_{2} = 1

By adding equation (2), (4) and (6), we get,

𝑦_{1 }+ 𝑦_{2} + 𝑦_{1} + 𝑦_{3} + 𝑦_{2} + 𝑦_{3} = −2 + 8 + 4

2𝑦_{1} + 2𝑦_{2} + 2𝑦_{3 }= −2 + 12

2(𝑦_{1} + 𝑦_{2} + 𝑦_{3}) = 10

𝑦_{1} + 𝑦_{2} + 𝑦_{3} = 5

Put the value of 𝑦_{1} + 𝑦_{2} in above equation,

−2 + 𝑦_{3} = 5

𝑦_{3} = 5 + 2

𝑦_{3} = 7

Put the value of 𝑦_{3 }in equation (4),

8 = 𝑦_{1} + 7

8 − 7 = 𝑦1

𝑦_{1} = 1

Put the value 𝑦1in equation (6),

−2 = 1 + 𝑦_{2}

−2 = 1 + 𝑦_{2}

−2 − 1 = 𝑦_{2}

−3 = 𝑦_{2}

𝑦_{2} = −3

**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. Calculates the values of x and y.

Solution:

It is given that, Points A (-5, x), B (y, 7) and C (1, -3) are collinear such that AB = BC

**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.

Solution:

It is given that, Points P (a, -4), Q (-2, b) and R (0, 2) are collinear. If Q lies between P and R, such that PR = 2QR.

According to question,

Q lies between P and R, PR=PQ+QR

2QR=PQ+QR

2QR-QR=PQ

QR=PQ

Q is the mid-point,

**Question **16. Calculate the co-ordinates of the centroid of a triangle ABC, if A = (7, -2), B = (0, 1) and C = (-1, 4).

Solution:

It is given that, co-ordinates of the centroid of a triangle ABC, if A = (7, -2), B = (0, 1) and C = (-1, 4).

**Question **17. The co-ordinates of the centroid of a PQR are (2, -5). If Q = (-6, 5) and R = (11, 8); calculate the co-ordinates of vertex P.

Solution:

Let us assumed that, G be the centroid of DPQR whose coordinates are (2, -5) and let (x,y) be the coordinates of vertex P.

**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.

Solution:

It is given that, the vertices of the triangle ABC whose centroid is the origin. The co-ordinates at the origin are (0,0).

### Exercise 13C

**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.**Solution:**

**Question **2. A (20, 0) and B (10, -20) are two fixed points. Find the co-ordinates of a point P in AB such that: 3PB = AB. Also, find the co-ordinates of some other point Q in AB such that AB = 6AQ.

Solution:

It is given that, 3PB=AB

The ratio is 1:5.

It is given that, co-ordinates of A (20, 0) and B (10, -20)

By using section formula

**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 = 3/8 BC.

Solution:

It is given that, the vertices of a triangle are A (-8, 0), B (0, 16) and C (0, 0) and the ratio of AP:PB are 3:5.

By using section formula,

Hence, the co-ordinates of Q are (-5,0).

Distance of PQ,

By distance formula,

We know that the distance formula,

**Question **4. Find the co-ordinates of points of trisection of the line segment joining the point (6, -9) and the origin.

Solution:

It is given that, the co-ordinates of point A (6,-9) and B (0,0)

Let us assumed that, Point P divides the line segment in the ratio 1:2 and co-ordinates of P (x, y).

By using section formula,

Hence, the co-ordinates of P (4,-6).

Let us assumed that, Point Q divides the line segment in the ratio 2:1 and co-ordinates of Q (x, y).

By using section formula,

**Question **5. A line segment joining A(-1, 5/3) and B (a, 5) is divided in the ratio 1: 3 at P, point where the line segment AB intersects the y-axis.**(i) Calculate the value of ‘a’.****(ii) Calculate the co-ordinates of ‘P’.Solution:**

It is given that, the line segment AB intersects the y-axis at point P

Let us assumed that, the co-ordinates of point P be (0, y) and P divides AB in the ratio 1: 3.

By using section formula,

**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.

Solution:

Let the line segment AB intersects the x-axis by point P(x,0) in the ratio K:1.

By using section formula,

**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.

Solution:

A lies on x-axis, co-ordinates of point A(x,0) and B lies on y-axis co-ordinates of point A(0,y).

By mid-point formula,

**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.Solution:**

**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)

Solution:

It is given that, the co-ordinates of the centroid of a triangle ABC vertices are A (-1, 3), B (1, -1) and C (5, 1).

**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.

Solution:

It is given that the mid-point of the line-segment joining (4a, 2b – 3) and (-4, 3b) is (2,-2a).

By mid-point formula,

**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.

Solution:

It is given that, the mid-point of (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?Solution:**

(i) It is given that, the co-ordinates of the point P that divides the line joining A (-4, 1) and B (17, 10) in the ratio 1: 2.

By Section formula,

**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.

Solution:

It is given that, the points A (-5, 4), B (-1, -2) and C (5, 2) are the vertices of an isosceles right- angled triangle.

104

𝐴𝐵 = 𝐵𝐶 so, it is a isosceles triangle

We know that, 𝐴𝐵^{2} + 𝐵𝐶^{2} = 𝐴𝐶^{2}

𝐴𝐵^{2} + 𝐵𝐶^{2} = 𝐴𝐶^{2}

Hence, it is an isosceles right-angles triangle.

If ABCD is a square,

Mid-point of AC = Mid-point of 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.

Solution:

It is given that, M is the mid-point of the line segment joining the points A (-3, 7) and B (9, -1).

The co-ordinates of point M are,