Question

Given: the coordinates of triangle PQR are (0,0), Q(2a,0), and R(2b,2c). Prove: the line containing the midpoints of two sides of a triangle is parallel to the third scale.
as part of the proof find the midpoint of QR


A) (a-b,c)
B) (a+b,c)
C) (a,c)
D) (b,c)

Answer 1

In your question where a triangle PQR having a coordinates of P(0,0), Q(2a,0), and R(2b,2c) to prove that the line containing the midpoint of two sides of a triangle is parallel to the third scale the midpoit of QR should be letter B. (a+b, c)

Answer 2

Answer: The midpoint of QR has co-ordinates $(a+b,c)$.

Step-by-step explanation: In ΔPQR, the coordinates of the vertices are $P(0,0),~Q(2a,0) ~\textup{and}~ R(2b,2c)$.

Let, S, T  and V be the mid-points of PQ, PS and QS respectively. Thus, the coordinates of S are $\left ( \frac{0+2a}{2},\frac{0+0}{2} \right )=(a,0)$

and coordinates of T are $\left ( \frac{0+2b}{2},\frac{0+2c}{2} \right )=(b,c)$.

Now, slope of line ST is $\frac{c-0}{b-a}=\frac{c}{b-a}$

and slope of QR is $\frac{2c-0}{2b-2a}=\frac{c}{b-a}$.

Since the slopes of ST and QR are equal, hence they must be parallel.

Also, coordinates of V are $\left ( \frac{2a+2b}{2},\frac{0+2c}{2} \right )=(a+b,c)$.