Question

Segment $s_1$ has endpoints at $(3+\sqrt{2},5)$ and$(4,7)$. Segment $s_2$ has endpoints at $(6-\sqrt{2},3)$ 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

Answer:

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

Step-by-step 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.

$x = \frac{3 + \sqrt{2} + 4}{2} = \frac{7 + \sqrt{2}}{2}$

$y = \frac{5 + 7}{2} = \frac{12}{2} = 6$

Thus:

$M_{s1} = (\frac{7 + \sqrt{2}}{2},6)$

Midpoint of s2:

$x = \frac{6 - \sqrt{2} + 3}{2} = \frac{9 - \sqrt{2}}{2}$

$y = \frac{3 + 5}{2} = \frac{8}{2} = 4$

Thus:

$M_{s2} = (\frac{9 - \sqrt{2}}{2}, 4)$

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

$x = \frac{\frac{7 + \sqrt{2}}{2} + \frac{9 - \sqrt{2}}{2}}{2} = \frac{16}{4} = 4$

$y = \frac{6 + 4}{2} = \frac{10}{2} = 5$

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