2. BY ASSOCIATE PROFESSOR NADEEM UDDIN
MIDPOINT
The midpoint K of the line segment connecting two points having coordinates (x1,y1) and (x2,y2)
has coordinates (
๐ฅ1+๐ฅ2
2
,
๐ฆ1+๐ฆ2
2
).
Example-8
Find the midpoint of the line segment connecting the following points
(1, 1) and (2, 5)
Solution:
๐ฅ1 = 1 ๐๐๐ ๐ฆ1 = 1 ; ๐ฅ2 = 2 ๐๐๐ ๐ฆ2 = 5
(
๐ฅ1 + ๐ฅ2
2
,
๐ฆ1 + ๐ฆ2
2
)
(
1 + 2
2
,
1 + 5
2
)
(
3
2
,
6
2
) = (
3
2
, 3)
Example-9
P(3/2, 3) is the mid-point of the join of A and B(- 2,5).Find the coordinates of A.
Solution: Let (๐ฅ1, ๐ฆ1) be the coordinate of A
We know that
x =
๐ฅ1+๐ฅ2
2
, y =
๐ฆ1+๐ฆ2
2
3
2
=
๐ฅ1โ 2
2
, 3 =
๐ฆ1+ 5
2
3
2
(2) = ๐ฅ1 โ 2 , 3(2) = ๐ฆ1 + 5
3 = ๐ฅ1 โ 2 , 6 = ๐ฆ1 + 5
๐ฅ1 = 3 + 2 = 5 , ๐ฆ1 = 6 โ 5 = 1
Hence coordinates of A is (5, 1)
3. DO YOURSELF
Find the midpoint of the line segment connecting the following points
i) (2, 2) and (6, 5) Ans (4, 7/2)
ii) (7, 8) and (2, 3) Ans (9/2, 11/2)
iii) (-2,-2) and (4, 6) Ans (1, 2)
iv) (10,7) and (12,-4) Ans (11, 3/2)
Example-10
Find the length and the mid-point of the line segment joining the points A(- 2, -2) and B(4, 6)
Solution:
๐ฅ1 = โ2 ๐๐๐ ๐ฆ1 = โ2 ; ๐ฅ2 = 4 ๐๐๐ ๐ฆ2 = 6
|๐ด๐ต| = โ(๐ฅ2 โ ๐ฅ1)2 + (๐ฆ2 โ ๐ฆ1)2
|๐ด๐ต| = โ(4 + 2)2 + (6 + 2)2
|๐ด๐ต| = โ(6)2 + (8)2
|๐ด๐ต| = โ36 + 64
|๐ด๐ต| = โ100
|๐ด๐ต| = 10 ๐ข๐๐๐ก๐
Mid-point = (
๐ฅ1+๐ฅ2
2
,
๐ฆ1+๐ฆ2
2
)
Mid-point = (
โ 2+4
2
,
โ 2+6
2
)
Mid-point = (1, 2)
4. Example-11
The end points of the diameter of a circle lie on the points (6, 5) and (3, 9).Find the centre and
radius of the circle.
Solution:
๐ฅ1 = 6 ๐๐๐ ๐ฆ1 = 5 ; ๐ฅ2 = 3 ๐๐๐ ๐ฆ2 = 9
Centre of the circle = Mid-point = (
๐ฅ1+๐ฅ2
2
,
๐ฆ1+๐ฆ2
2
)
Centre of the circle = Mid-point = (
6+3
2
,
5+9
2
)
Centre of the circle = Mid-point = (
9
2
, 7)
๐ ๐๐๐๐ข๐ of the circle = ๐ท = โ(๐ฅ2 โ ๐ฅ1)2 + (๐ฆ2 โ ๐ฆ1)2
๐ ๐๐๐๐ข๐ of the circle = ๐ท = โ(3 โ 6)2 + (9 โ 5)2
๐ ๐๐๐๐ข๐ of the circle = ๐ท = โ(โ3)2 + (4)2
๐ ๐๐๐๐ข๐ of the circle = ๐ท = โ9 + 16
๐ ๐๐๐๐ข๐ of the circle = ๐ท = โ25
๐ ๐๐๐๐ข๐ of the circle = ๐ท = 5 ๐ข๐๐๐ก๐