If the coordinates of the midpoint of the line segment with end points (a, 4) and (3, b) are (5, -2), find the values of a and b.
(a + 3) / 2 = 5
(b + 4) / 2 = -2
(a,4), (5,-2), (3,b).
5-a = (3-a)/2.
10-2a = 3 - a,
a = 7.
-2-4 = (b-4)/2.
To find the values of a and b, we can use the midpoint formula. The midpoint formula states that the coordinates of the midpoint between two points (x₁, y₁) and (x₂, y₂) is:
((x₁ + x₂) / 2, (y₁ + y₂) / 2)
In this problem, we are given the following information:
Midpoint coordinates: (5, -2)
Endpoint coordinates: (a, 4) and (3, b)
Using the midpoint formula, we can set up the following equations:
(5, -2) = ((a + 3) / 2, (4 + b) / 2)
To solve for a and b, we need to isolate a and b in the equation:
(5, -2) = ((a + 3) / 2, (4 + b) / 2)
Multiplying both sides of the equation by 2, we get:
(10, -4) = (a + 3, 4 + b)
From the equation, we can see that a + 3 = 10 and 4 + b = -4. Solving these equations, we find:
a = 10 - 3 = 7
b = -4 - 4 = -8
Therefore, the values of a and b are 7 and -8, respectively.