The points t(2, 1), u(4, 5), and v(7, 4) are the midpoints of the sides of triangle abc. graph the three midsegments. complete the coordinates of each vertex of triangle abc.

In the attached diagram, we have arbitrarily assigned A, B, C to be opposite the sides having midpoints T, U, V, respectively.

The coordinates of A are the reflection of midpoint T across the midpoint of segment UV, so can be computed as ...

... A = U + V - T = (4, 5) +(7, 4) -(2, 1) = (4+7-2, 5+4-1) = (9, 8)

Likewise, ...

... B = V + T - U = (7+2-4, 4+1-5) = (5, 0)

... C = T + U - V = (2+4-7, 1+5-4) = (-1, 2)

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Reflection across a point

Suppose we want point M to be the midpoint between A and B. That is, B is the reflection of point A across point M. Then the required relationship is ...

... M = (A+B)/2

Solving for B, we find ...

... 2M = A+B

... 2M -A = B

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Reflection across the midpoint of a segment

Suppose we have segment PQ and point A and we want to reflect point A across the midpoint of PQ to give point B. Let M be the midpoint of PQ. Then ...

... M = (P+Q)/2

We know from above that ...

... B = 2M -A

Substituting for M, we have ...

... B = 2(P+Q)/2 -A

... B = P + Q - A . . . . . . . the formulation we used in the solution above.

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Further comment on reflection across midpoints

You will occasionally see a problem where you are to find the 4th vertex of a rectangle or parallelogram, given three vertices. That problem is essentially the problem of reflecting one vertex across the midpoint of the diagonal between the other two given vertices. The calculation is the same as that described above.

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