Answer and Step-by-step explanation:
The coordinates of C are (5 , 2)
The slope of CD is 3
The coordinates of D are (6 , 5) and (4 , -1)
Step-by-step explanation:
* Now lets study the problem
- The ends points of line AB are A = 2 , 3) and B = (8 , 1)
- CD is the perpendicular bisector of AB, and C lies on AB
- That means:
# C is the mid-point of AB
# The slope of AB Γ the slope of CD = -1 (one of them is a multiplicative
Β inverse and additive inverse of the other)
-Ex: the slope of one is a/b, then the slope of the other is -b/a
* The mid-point between two points (x1 , y1) and (x2 , y2) is:
Β [(x1 + x2)/2 , (y1 + y2)/2]
β΅ C is the mid-point of AB
β΄ C = [(2 + 8)/2 , (3 + 1)/2] = [10/2 , 4/2] = (5 , 2)
* The coordinates of C are (5 , 2)
- The slope of a line passing through points (x1 , y1) and (x2 , y2) is:
the slope = (y2 - y1)/(x2 - x1)
β΄ The slope of AB = (1 - 3)/(8 -2) = -2/6 = -1/3
β΅ CD β₯ AB
β΄ The slope of CD Γ the slope of AB = -1
β΄ The slope of CD = 3
* The slope of CD is 3
- The length of a line passing through points (x1 , y1) and (x2 , y2) is:
the length = β[(x2 - x1)Β² + (y2 - y1)Β²]
β΅ The length of CD = β10
β΅ Point D is (x , y)
β΄ (x - 5)Β² + (y - 2)Β² = (β10)Β²
β΄ (x - 5)Β² + (y - 2)Β² = 10 β (1)
β΅ The slope of CD is (y - 2)/(x - 5) = 3 β by using cross multiply
β΄ (y - 2) = 3(x - 5) β (2)
- Substitute (2) in (1)
β΄ (x - 5)Β² + [3(x - 5)]Β² = 10 β simplify
* [3(x - 5)]Β² = (3)Β²(x - 5)Β² = 9(x - 5)Β²
β΄ (x - 5)Β² + 9(x - 5)Β² = 10 β add the like terms
β΄ 10(x - 5)Β² = 10 β Γ· 10 both sides
β΄ (x - 5)Β² = 1 β take β for both sides
β΄ x - 5 = Β± 1
β΄ x - 5 = 1 β add 5 to both sides
β΄ x = 6
* OR
β΄ x - 5 = -1 β add 5 to both sides
β΄ x = 4
- Substitute the values of x in (2)
β΄ y - 2 = 3(6 - 5)
β΄ y - 2 = 3 β add 2
β΄ y = 5
* OR
β΄ y - 2 = 3(4 - 5)
β΄ y - 2 = -3 β add 2
β΄ y = -1
* The coordinates of D are (6 , 5) and (4 , -1)