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 (
𝑥1+𝑥2
2
,
𝑦1+𝑦2
2
).
Example-8
Find the midpoint of the line segment connecting the following points
(1, 1) and (2, 5)
Solution:
𝑥1 = 1 𝑎𝑛𝑑 𝑦1 = 1 ; 𝑥2 = 2 𝑎𝑛𝑑 𝑦2 = 5
(
𝑥1 + 𝑥2
2
,
𝑦1 + 𝑦2
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 (𝑥1, 𝑦1) be the coordinate of A
We know that
x =
𝑥1+𝑥2
2
, y =
𝑦1+𝑦2
2
3
2
=
𝑥1− 2
2
, 3 =
𝑦1+ 5
2
3
2
(2) = 𝑥1 − 2 , 3(2) = 𝑦1 + 5
3 = 𝑥1 − 2 , 6 = 𝑦1 + 5
𝑥1 = 3 + 2 = 5 , 𝑦1 = 6 − 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:
𝑥1 = −2 𝑎𝑛𝑑 𝑦1 = −2 ; 𝑥2 = 4 𝑎𝑛𝑑 𝑦2 = 6
|𝐴𝐵| = √(𝑥2 − 𝑥1)2 + (𝑦2 − 𝑦1)2
|𝐴𝐵| = √(4 + 2)2 + (6 + 2)2
|𝐴𝐵| = √(6)2 + (8)2
|𝐴𝐵| = √36 + 64
|𝐴𝐵| = √100
|𝐴𝐵| = 10 𝑢𝑛𝑖𝑡𝑠
Mid-point = (
𝑥1+𝑥2
2
,
𝑦1+𝑦2
2
)
Mid-point = (
− 2+4
2
,
− 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:
𝑥1 = 6 𝑎𝑛𝑑 𝑦1 = 5 ; 𝑥2 = 3 𝑎𝑛𝑑 𝑦2 = 9
Centre of the circle = Mid-point = (
𝑥1+𝑥2
2
,
𝑦1+𝑦2
2
)
Centre of the circle = Mid-point = (
6+3
2
,
5+9
2
)
Centre of the circle = Mid-point = (
9
2
, 7)
𝑅𝑎𝑑𝑖𝑢𝑠 of the circle = 𝐷 = √(𝑥2 − 𝑥1)2 + (𝑦2 − 𝑦1)2
𝑅𝑎𝑑𝑖𝑢𝑠 of the circle = 𝐷 = √(3 − 6)2 + (9 − 5)2
𝑅𝑎𝑑𝑖𝑢𝑠 of the circle = 𝐷 = √(−3)2 + (4)2
𝑅𝑎𝑑𝑖𝑢𝑠 of the circle = 𝐷 = √9 + 16
𝑅𝑎𝑑𝑖𝑢𝑠 of the circle = 𝐷 = √25
𝑅𝑎𝑑𝑖𝑢𝑠 of the circle = 𝐷 = 5 𝑢𝑛𝑖𝑡𝑠
