Answer:
The Coordinates of Point C are C(5,2)
The slope of CD is 3
The possible coordinates of point D(a,b) are (6, 5) and (4, -1)
Step-by-step explanation:
According to the Question,
- Given, The end points of AB are A(2,3) and B(8,1). The perpendicular Bisector of AB is CD, and point C lies on AB. The length of the CD is √10 units.
Now, Let The Coordinates of Point C are C( x , y ) .
- Thus, C( x , y ) = ((8 +2) / 2 , ( 1 + 3 )/ 2 ) ⇒ (10/2 , 4/2) ⇒ (5 , 2)
The Coordinates of Point C are C( x , y ) ⇒C(5, 2).
And, The slope of AB = (1 - 3)/(8 - 2) ⇒ -2/6 ⇔ -1/3 .
- Thus, The slope of CD is -1/(The slope of AB) = -1/(-1/3) ⇔ 3.
Let the coordinate of D be (a, b) then
⇒ √{ (b - 2)² +(a - 5)² } = √10
on squaring both sides we get,
⇒ a² - 10a + 25 + b² - 4b + 4 = 10
⇒ a² + b² - 10a - 4b = - 19 ⇔⇔ (Equation 1)
We Know, the slope of CD = 3
⇒Thus, (b - 2)/(a - 5) = 3
⇒ b - 2 = 3a - 15
⇒ b = 3a - 13 ⇔⇔ (Equation 2)
Putting value of Equation 2 into Equation 1, We get
⇒a² + (3a - 13)² - 10a - 4(3a - 13) = - 19
⇒a² + 9a² - 78a + 169 - 10a - 12a + 52 = - 19
⇒10a² - 100a + 240 = 0
⇒a² - 10a + 24 = 0
⇒(a - 4)(a - 6) = 0
⇒a = 4 or a = 6
Now,
When a = 4 , b = 3(4) - 13 ⇒ 12 - 13 ⇒ b = -1
When a = 6, b = 3(6) - 13 ⇒ 18 - 13 ⇒ b = 5
Therefore, the possible coordinates of point D(a,b) are (6, 5) and (4, -1).
