**Find the Midpoint of the Line Segment Whose Endpoints are Given**

Here we are going to see some example problems to find the midpoint of the line segment when endpoints are given.

The mid-point M of the line segment joining the points A(x_{1}, y_{1}) and B(x_{2} , y_{2}) is

**Question 1 :**

Find the mid-points of the line segment joining the points

(i) (−2, 3) and (−6,−5)

**Solution :**

(x_{1}, y_{1}) ==> (-2, 3)

(x_{2} , y_{2}) ==> (-6, -5)

Midpoint = (-2 - 6)/2, (3 + (-5))/2

= -8/2, -2/2

= (-4, -1)

(ii) (8,−2) and (−8,0)

**Solution :**

(x_{1}, y_{1}) ==> (8, -2)

(x_{2} , y_{2}) ==> (-8, 0)

Midpoint = (8 - 8)/2, (-2 + 0)/2

= 0/2, -2/2

= (0, -1)

(iii) (a, b) and (a + 2b, 2a - b)

**Solution :**

(x_{1}, y_{1}) ==> (a, b)

(x_{2} , y_{2}) ==> (a + 2b, 2a - b)

Midpoint = (a + a + 2b)/2, (b + 2a - b)/2

= 2(a + b)/2, 2a/2

= (a + b, 1)

(iv) (1/2, -3/7) and (3/2, -11/7)

**Solution :**

(x_{1}, y_{1}) ==> (1/2, -3/7)

(x_{2} , y_{2}) ==> (3/2, -11/7)

Midpoint = ((1/2) + (3/2))/2, ((-3/7) + (-11/7))/2

= ((4/2)/2, ((-14/7)/2)

= (1, -1)

**Question 2 :**

The centre of a circle is (−4,2). If one end of the diameter of the circle is (−3,7), then find the other end

**Solution :**

Midpoint of the diameter = Center of the circle

Let the other endpoint be (a, b)

Midpoint of (-3, 7) and (a, b) is (-4, 2).

(-3 + a)/2, (7 + b)/2 = (-4, 2)

By equating the x and y coordinates, we get

(-3 + a)/2 = -4 -3 + a = -8 a = -8 + 3 a = -5 |
(7 + b)/2 = 2 7 + b = 4 b = 4 - 7 b = -3 |

Hence the other end is (-5, -3).

**Question 3 :**

If the mid-point (x, y) of the line joining (3, 4) and (p, 7) lies on 2x + 2y + 1 = 0 , then what will be the value of p?

**Solution :**

Midpoint of the line segment joining the points (3, 4) and (p, 7)

(3 + p)/2 , (7 + 4)/2 = (x, y)

(3 + p)/2 , 11/2 = (x, y)

x = (3 + p)/2 and y = 11/2

Since the midpoint lies on the line 2x + 2y + 1 = 0

2(3 + p)/2 + 2(11/2) + 1 = 0

3 + p + 11 + 1 = 0

p + 15 = 0

p = -15

After having gone through the stuff given above, we hope that the students would have understood, "Find the Midpoint of the Line Segment Whose Endpoints are Given"

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