In triangle GHJ, K(2,3) is the midpoint of segment GH, L(4,1) is the midpoint of segment HJ, and M(6,2) is the midpoint of segment GJ. What are the coordinates of G, H, and J?
For this problem, I doubled each of the coordinates because of the Triangle Midpoint Theorem. The problem is that I do not which coordinate belongs to which point. Was I correct in doubling the coordinates (i.e. (4,1) → (8, 2))?
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After drawing the triangle KLM, we need to apply some geometric facts. They are:-
(1) Triangle GHJ will have sides parallel to the sides of KLM. For example, GLJ // KM.
(2) Then, HKLM forms a parallelogram with diagonals intersecting at X.
(3) Hence, by the properties of a parallelogram, X is the midpoint of KM.
From (3), we can use the midpoint formula to find the co-ordinates of X.
Since X is also the midpoint of HL, we can then use the same formula to find the co-ordinates of H.
Those of G and J can be found in the similar way.
The midpoint of a line segment with endpoints $(a_1,b_1)$ and $(a_2,b_2)$ is $$\left(\frac{a_1+a_2}{2},\frac{b_1+b_2}{2}\right).$$ For your case, we define the vertices as $G(x_1,y_1)$, $H(x_2,y_2)$, and $J(x_3,y_3)$. Consequently, $$\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)=\left(2,3\right),\tag 1$$ $$\left(\frac{x_2+x_3}{2},\frac{y_2+y_3}{2}\right)=\left(4,1\right),\tag 2$$ and $$\left(\frac{x_3+x_1}{2},\frac{y_3+y_1}{2}\right)=\left(6,2\right).\tag 3$$
Adding $(1)$, $(2)$, and $(3)$ results in $$x_1+x_2+x_3=12, \tag 4$$ and $$y_1+y_2+y_3=6. \tag 5$$ From $(1)$, $x_1+x_2 = 4 \implies x_1+x_2 + x_3 = 4+x_3 = 12 \implies x_3 = 8$.
We can solve the remaining unknowns similarly. Your final answer should be $(x_1,y_1)=(4,4)$, $(x_2,y_2)=(0,2)$, and $(x_3,y_3)=(8,0)$.