For #15 & 18, the x-coordinate for the midpoint can be found by adding the x-values and dividing by 2. the y-coordinate of the midpoint can be found by add the y-values and dividing by 2.
I will do #15 as an example:
(3, -5) and (7, 9)
M = ([tex] \frac{x1 + x2}{2} [/tex], [tex] \frac{y1 + y2}{2} [/tex])
= ([tex] \frac{3 + 7}{2} [/tex], [tex] \frac{-5 + 9}{2} [/tex])
= ([tex] \frac{10}{2} [/tex], [tex] \frac{4}{2} [/tex])
= (5, 2)
#19) use the same concept as in #15, however you have the midpoint and one of the endpoints so split the x's and y's into two separate equations.
endpoint is (5, -6), midpoint is (4, 3)
xM = [tex] \frac{x1 + x2}{2} [/tex] yM = [tex] \frac{y1 + y2}{2} [/tex]
Multiply both sides of both equations by 2 to eliminate the denominator
2(xM) = x₁ + x₂ 2(yM) = y₁ + y₂
Now, plug in the midpoint and endpoint values and then solve.
2(4) = 5 + x₂ 2(3) = -6 + y₂
8 = 5 + x₂ 6 = -6 + y₂
3 = x₂ 12 = y₂
Endpoint coordinate is (3, 12)
For #23 & 24, use the distance formula: d = [tex] \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}} [/tex]
I will do #23 as an example: (13,2) and (7, 10)
d = [tex] \sqrt{(7 - 13)^{2} + (10 - 2)^{2}} [/tex]
= [tex] \sqrt{(-6)^{2} + (8)^{2}} [/tex]
= [tex] \sqrt{36 + 64} [/tex]
= [tex] \sqrt{100} [/tex]
= 10
