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

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 ( (0+2a)/(2),(0+0)/(2) \right )=(a,0)`

and coordinates of T are
`\left ( (0+2b)/(2),(0+2c)/(2) \right )=(b,c)`.

Now, slope of line ST is
`(c-0)/(b-a)=(c)/(b-a)`

and slope of QR is
`(2c-0)/(2b-2a)=(c)/(b-a)`.

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

Also, coordinates of V are
`\left ( (2a+2b)/(2),(0+2c)/(2) \right )=(a+b,c)`.

Answer 2

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)