Question

I need help on how to simplify the numerators of the coordinates for this equation. Please explain all on how to get the answer. Thank you

Answer 1

`~~~~~~~~~~~~\textit{middle point of 2 points } \\\\ (\stackrel{x_1}{(3)/(8)}~,~\stackrel{y_1}{-(1)/(3)})\qquad (\stackrel{x_2}{-(1)/(2)}~,~\stackrel{y_2}{(5)/(8)}) \qquad \left(\cfrac{ x_2 + x_1}{2}~~~ ,~~~ \cfrac{ y_2 + y_1}{2} \right) \\\\\\ \left(\cfrac{~~ -( 1 )/( 2 )+(3)/(8) ~~}{2}~~,~~\cfrac{~~ ( 5 )/( 8 )-(1)/(3) ~~}{2} \right)\implies \left(\cfrac{~~ ((4)(-1)+(1)(3))/(8) ~~}{2}~~,~~\cfrac{~~ ((3)(5)+(8)(-1))/(24) ~~}{2} \right)`

`\left(\cfrac{~~ (-4+3)/(8) ~~}{2}~~,~~\cfrac{~~ (15-8)/(24) ~~}{2} \right)\implies \left(\cfrac{~~ (-1)/(8) ~~}{2}~~,~~\cfrac{~~ (7)/(24) ~~}{2} \right) \implies \left(\cfrac{~~ (-1)/(8) ~~}{(2)/(1)}~~,~~\cfrac{~~ (7)/(24) ~~}{(2)/(1)} \right) \\\\\\ \left(\cfrac{-1}{8}\cdot \cfrac{1}{2}~~,~~\cfrac{7}{24}\cdot \cfrac{1}{2} \right)\implies \left(-\cfrac{1}{16}~~,~~\cfrac{7}{48} \right)`

Answer 2

(-1/16, 7/48)

Explanation:

You want the midpoint of the segment with endpoint coordinates (3/8, -1/3) and (-1/2, 5/8).

Midpoint

The midpoint is the average of the endpoint coordinates:

M = ((3/8, -1/3) +(-1/2, 5/8))/2

M = (3/8 -1/2, -1/3 +5/8)/2 = (3/8 -4/8, (-8 +15)/24)/2

M = (-1/8, 7/24)/2 = (-1/16, 7/48)

The midpoint of the segment is (-1/16, 7/48).

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Additional comment

You recognize that 1/2 and 3/8 can use the denominator 8 for both fractions. The fractions 1/3 and 5/8 will need a common denominator of 24.

Here, the numerators of the sums are odd, so cannot be divided by 2. Instead, we multiply the denominator by 2 to divide the fraction by 2.

For the y-coordinates, we used this fraction sum:

a/b + c/d = (ad +bc)/(bd)

This always works, so can be a way to add fractions without concern for a common denominator. In the end, you may have to reduce the fraction. (Here, we do not have to reduce the result.)

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