Question

Segment
`$s_1$` has endpoints at
`$(3+√(2),5)$` and
`$(4,7)$`. Segment
`$s_2$` has endpoints at
`$(6-√(2),3)$` and
`$(3,5)$`. Find the midpoint of the segment with endpoints at the midpoints of
`$s_1$` and
`$s_2$`. Express your answer as
`$(a,b)$`.

Answer 1

The midpoint of the segment with endpoints at the midpoints of s1 and s2 is (4,5).

Explanation:

Midpoint of a segment:

The coordinates of the midpoint of a segment are the mean of the coordinates of the endpoints of the segment.

Midpoint of s1:

Using the endpoints given in the exercise.

`x = (3 + √(2) + 4)/(2) = (7 + √(2))/(2)`

`y = (5 + 7)/(2) = (12)/(2) = 6`

Thus:

`M_(s1) = ((7 + √(2))/(2),6)`

Midpoint of s2:

`x = (6 - √(2) + 3)/(2) = (9 - √(2))/(2)`

`y = (3 + 5)/(2) = (8)/(2) = 4`

Thus:

`M_(s2) = ((9 - √(2))/(2), 4)`

Find the midpoint of the segment with endpoints at the midpoints of s1 and s2.

Now the midpoint of the segment with endpoints
`M_(s1)` and
`M_(s2)`. So

`x = ((7 + √(2))/(2) + (9 - √(2))/(2))/(2) = (16)/(4) = 4`

`y = (6 + 4)/(2) = (10)/(2) = 5`

The midpoint of the segment with endpoints at the midpoints of s1 and s2 is (4,5).