Question 515768: Hello. I have a problem saying that the points T(2,1), U(4,5), and V(7,4) are all the midpoints of a triangle. The first thing that i have to do is that i have to plot the midsegments of that triangle with those midpoints, which i understand how to do (you just plot those points and connect them). Then the problem tells me to draw the original triangle and give the coordinates of each vertex. I have already drawn the original triangle because i know that the original sides are parallel to the midsegments, but i do not understand how to find the coordinates of each vertex. Could you please help me with this?
Answer by drcole(72)   (Show Source):
You can put this solution on YOUR website! Let triangle ABC be the original triangle. Let T be the midpoint of AB, let U be the midpoint of BC, and let V be the midpoint of CA. We are given the coordinates T(2,1), U(4,5) and V(7,4). You are correct that the midsegment TU is parallel to CA, UV is parallel to AB, and VT is parallel to BC. Another useful fact is that the length of each midsegment is equal to one half the length of the side of the triangle to which it is parallel. You can prove this using similar triangle. So:
the length of TU is one half the length of CA;
the length of UV is one half the length of AB; and
the length of VT is one half the length of BC.
Another way to say this is that, since T, U, and V are the midpoints of AB, BC, and CA respectively,
the vector TU (the directed line segment from T to U) has the same length and direction as the vector VC;
the vector UV has the same length and direction as the vector TA; and
the vector VT has the same length and direction as the vector UB.
This will help us find the coordinates of A, B, and C. First, let's find the vector TU:
vector TU = "U - T" = <4 - 2, 5 - 1> = <2, 4> (in other words, to go from T to U, we move right two units, and up four units).
Now we know that the vector VC has the same length and direction as the vector TU, so to from V to C, we should also move right two units and up four units. So, since the coordinates of V are (7,4), we get:
C = (7 + 2, 4 + 4) = (9, 8).
We can do the same to find A and B. The vector UV will equal:
vector UV = "V - U" = <7 - 4, 4 - 5> = <3, -1> (to go from U to V, we move right three units and down one unit).
Since T = (2, 1), we get:
A = (2 + 3, 1 + (-1)) = (5, 0).
Finally, the vector VT will equal:
vector VT = "T - V" = <2 - 7, 1 - 4> = <-5, -3> (to go from V to T, we move left five units and down three units).
Since U = (4, 5), we get:
B = (4 + (-5), 5 + (-3)) = (-1, 2).
So the vertices of the original triangle are at A(5, 0), B(-1, 2), and C(9, 8). Your drawing should confirm this. You can also check this answer by noticing that the midpoint of AB is indeed T(2, 1), the midpoint of BC is indeed U(4, 5), and the midpoint of CA is indeed V(7, 4), which could only occur if the coordinates we have for A, B, and C are the correct coordinates for the original triangle.
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