Place a triangle in the coordinate plane. Use the midpoint formula to find the coordinates of the midpoints and then find the slope of the relevant segments.
See solution.
Practice makes perfect
Let's consider a triangle ABC with vertices A(0,0), B(a,0), and C(b,c). Also, let S and T be the midpoints of AC and BC, respectively.
Our mission is to prove that ST is parallel to AB. Before drawing ST, let's find their coordinates by using the Midpoint Formula.
M(x_1+x_2/2,y_1+y_2/2)
A(0,0) and C(b,c)
B(a,0) and C(b,c)
S(0+b/2,0+c/2) or S(b/2,c/2)
T(a+b/2,0+c/2) or T(a+b/2,c/2)
Next, let's connect S and T.
Our next step is to find the slope of the two relevant segments.
m = y_2-y_1/x_2-x_1
Let's find the slope of AB.
Because both segments have the same slope, we conclude that they are parallel: ST∥ AB.
Coordinate Proof
Given: & â–³ ABC
& S is the midpoint ofAC
& T is the midpoint ofBC
Prove: & ST∥ AB
Proof: Let A(0,0), B(a,0), and C(b,c) be the vertices of â–³ ABC. The Midpoint Formula allows us to find the coordinates of S and T, which are ( b2, c2) and ( a+b2, c2), respectively. Next, we calculate the slope of ST which is given by c/2-c/2(a+b)/2-b/2 or 0. Lastly, the slope of AB is 0-0a-0=0.
Because the slope of ST and AB are equal, we conclude that ST∥ AB.
