**Question 1.Find the co-ordinates of point P if P divides the line segment joining the points A (-1, 7) and B (4, -3) in the ratio 2:3.Solution:**

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

_{1}, y

_{1}) B (x

_{2}, y

_{2}) be the given points.

Here, x

_{1}= -1, y

_{1}= 7, x

_{2}= 4, y

_{2}= -3, m = 2, n = 3

∴ By section formula,

∴ The co-ordinates of point P are (1,3).

**Question 2.In each of the following examples find the co-ordinates of point A which divides segment PQ in the ratio a : b.i. P (-3, 7), Q (1, -4), a : b = 2 : 1ii. P (-2, -5), Q (4, 3), a : b = 3 : 4iii. P (2, 6), Q (-4, 1), a : b = 1 : 2Solution:**

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

i. Let P (x

_{1}, y

_{1}), Q (x

_{2}, y

_{2}) be the given points.

Here, x

_{1}= -3, y

_{1}= 7, x

_{2}= 1, y

_{2}= -4, a = 2, b = 1

∴ By section formula,

ii. Let P (x_{1},y_{1}), Q (x_{2}, y_{2}) be the given points.

Here, x_{1} = -2, y_{1} = -5, x_{2} = 4, y_{2} = 3, a = 3, b = 4

By section formula,

iii. Let P (x_{1}, y_{1}), Q (x_{2}, y_{2}) be the given points.

Here,x_{1} = 2,y_{1} = 6, x_{2} = -4, y_{2} = 1, a = 1,b = 2

∴ By section formula,

**Question 3.Find the ratio in which point T (-1, 6) divides the line segment joining the points P (-3,10) and Q (6, -8).Solution:**

Let P (x

_{1}, y

_{1}), Q (x

_{2}, y

_{2}) and T (x, y) be the given points.

Here, x

_{1}= -3, y

_{1}= 10, x

_{2}= 6, y

_{2}= -8, x = -1, y = 6

∴ By section formula,

∴ Point T divides seg PQ in the ratio 2 : 7.

**Question 4.Point P is the centre of the circle and AB is a diameter. Find the co-ordinates of point B if co-ordinates of point A and P are (2, -3) and (-2,0) respectively.Solution:**Let A (x

_{1}, y

_{1}), B (x

_{2}, y

_{2}) and P (x, y) be the given points.

Here, x

_{1}= 2, y

_{1}=-3,

x = -2, y = 0

Point P is the midpoint of seg AB.

∴ By midpoint formula,

∴ The co-ordinates of point B are (-6,3).

**Question 5.Find the ratio in which point P (k, 7) divides the segment joining A (8, 9) and B (1,2). Also find k.Solution:**

Let A (x

_{1}, y

_{1}), B (x

_{2}, y

_{2}) and P (x, y) be the given points.

Here, x

_{1}= 8, y

_{1}= 9, x

_{2}= 1, y

_{2}= 2, x = k, y = 7

∴ By section formula,

∴ Point P divides seg AB in the ratio 2 : 5, and the value of k is 6.

**Question 6.Find the co-ordinates of midpoint of the segment joining the points (22, 20) and (0,16).Solution:**

Let A (x

_{1}, y

_{1}) = A (22, 20),

B (x

_{2},y

_{2}) = B (0, 16)

Let the co-ordinates of the midpoint be P (x,y).

∴ By midpoint formula,

The co-ordinates of the midpoint of the segment joining (22, 20) and (0, 16) are (11,18).

**Question 7.Find the centroids of the triangles whose vertices are given below.i. (-7, 6), (2,-2), (8, 5)ii. (3, -5), (4, 3), (11,-4)iii. (4, 7), (8, 4), (7, 11)Solution:**

i. Let A (x

_{1}, y

_{1}) = A (-7, 6),

B (x

_{2}, y

_{2}) = B (2, -2),

C (x

_{3}, y

_{3}) = C(8, 5)

∴ By centroid formula,

∴ The co-ordinates of the centroid are (1,3).

ii. Let A (x_{1} y_{1}) = A (3, -5),

B (x_{2}, y_{2}) = B (4, 3),

C(x_{3}, y_{3}) = C(11,-4)

∴ By centroid formula,

∴ The co-ordinates of the centroid are (6, -2).

iii. Let A (x_{1}, y_{1}) = A (4, 7),

B (x_{2}, y_{2}) = B (8,4),

C (x_{3}, y_{3}) = C(7,11)

∴ By centroid formula,

**Question 8.In ∆ABC, G (-4, -7) is the centroid. If A (-14, -19) and B (3, 5), then find the co-ordinates of C.Solution:**G (x, y) = G (-4, -7),

A (x

_{1}, y

_{1}) = A (-14, -19),

B(x

_{2}, y

_{2}) = B(3,5)

Let the co-ordinates of point C be (x

_{3}, y

_{3}).

G is the centroid.

By centroid formula,

∴ The co-ordinates of point C are (-1, – 7).

**Question 9.A (h, -6), B (2, 3) and C (-6, k) are the co-ordinates of vertices of a triangle whose centroid is G (1,5). Find h and k.Solution:**A(x

_{1},y

_{1}) = A(h, -6),

B (x

_{2}, y

_{2}) = B(2, 3),

C (x

_{3}, y

_{3}) = C (-6, k)

∴ centroid G (x, y) = G (1, 5)

G is the centroid.

By centroid formula,

∴ 3 = h – 4

∴ h = 3 + 4

∴ h = 7

∴ 15 = -3 + k

∴ k = 15 + 3

∴ k = 18

∴ h = 7 and k = 18

**Question 10.Find the co-ordinates of the points of trisection of the line segment AB with A (2,7) and B (-4, -8).Solution:**A (2, 7), B H,-8)

Suppose the points P and Q trisect seg AB.

∴ AP = PQ = QB

∴ Point P divides seg AB in the ratio 1:2.

∴ By section formula,

Co-ordinates of P are (0, 2).

Point Q is the midpoint of PB.

By midpoint formula,

Co-ordinates of Q are (-2, -3).

∴ The co-ordinates of the points of trisection seg AB are (0,2) and (-2, -3).

**Question 11.If A (-14, -10), B (6, -2) are given, find the co-ordinates of the points which divide segment AB into four equal parts.Solution:**

Let the points C, D and E divide seg AB in four equal parts.

Point D is the midpoint of seg AB.

∴ By midpoint formula,

∴ Co-ordinates of D are (-4, -6).

Point C is the midpoint of seg AD.

∴ By midpoint formula,

∴ Co-ordinates of C are (-9, -8).

Point E is the midpoint of seg DB.

∴ By midpoint formula,

∴ Co-ordinates of E are (1,-4).

∴ The co-ordinates of the points dividing seg AB in four equal parts are C(-9, -8), D(-4, -6) and E(1, – 4).

**Question 12.If A (20, 10), B (0, 20) are given, find the co-ordinates of the points which divide segment AB into five congruent parts.Solution:**

Suppose the points C, D, E and F divide seg AB in five congruent parts.

∴ AC = CD = DE = EF = FB

∴ co-ordinates of C are (16, 12).

E is the midpoint of seg CB.

By midpoint formula,

∴ co-ordinates of E are (8, 16).

D is the midpoint of seg CE.

∴ co-ordinates of F are (4, 18).

∴ The co-ordinates of the points dividing seg AB in five congruent parts are C (16, 12), D (12, 14), E (8, 16) and F (4, 18).

**Intext Questions and Activities**

**Question 1.A (15, 5), B (9, 20) and A-P-B. Find the ratio in which point P (11, 15) divides segment AB. Find the ratio using x and y co-ordinates. Write the conclusion. (Textbook pg. no. 113)Solution:**

Suppose point P (11,15) divides segment AB in the ratio m : n.

By section formula,

∴ Point P divides seg AB in the ratio 2 : 1.

The ratio obtained by using x and y co-ordinates is the same.

**Question 2.External division: (Textbook pg. no. 115)Suppose point R divides seg PQ externally in the ratio 3:1.**

Let the common multiple be k.

Let PR = 3k and QR = k

Now, PR = PQ + QR … [P – Q – R]

∴ 3k = PQ + k

∴ Point Q divides seg PR in the ratio 2 : 1 internally.

Thus, we can find the co-ordinates of point R, when co-ordinates of points P and Q are given.