Question

Given: The coordinates of triangle PQR are P(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 side.

As part of the proof, find the midpoint of PR

Answer 1

b,c

Explanation:

That guy above took so long

Answer 2

The line containing the midpoints of two sides of a triangle is parallel to the third side ⇒ proved down

Explanation:

* Lets revise the rules of the midpoint and the slope to prove the

problem

- The slope of a line whose endpoints are (x1 , y1) and (x2 , y2) is

`m=(y_(2)-y_(1))/(x_(2)-x_(1))`

- The mid-point of a line whose endpoints are (x1 , y1) and (x2 , y2) is

`((x_(1)+x_(2))/(2),(y_(1)+y_(2))/(2))`

* Lets solve the problem

- PQR is a triangle of vertices P (0 , 0) , Q (2a , 0) , R (2b , 2c)

- Lets find the mid-poits of PQ called A

∵ Point P is (x1 , y1) and point Q is (x2 , y2)

∴ x1 = 0 , x2 = 2a and y1 = 0 , y2 = 0

∵ A is the mid-point of PQ

∴
`A=((0+2a)/(2),(0+0)/(2))=((2a)/(2),(0)/(2))=(a,0)`

- Lets find the mid-poits of PR which called B

∵ Point P is (x1 , y1) and point R is (x2 , y2)

∴ x1 = 0 , x2 = 2b and y1 = 0 , y2 = 2c

∵ B is the mid-point of PR

∴
`B=((0+2b)/(2),(0+2c)/(2))=((2b)/(2),(2c)/(2))=(b,c)`

- The parallel line have equal slopes, so lets find the slopes of AB and

QR to prove that they have same slopes then they are parallel

# Slope of AB

∵ Point A is (x1 , y1) and point B is (x2 , y2)

∵ Point A = (a , 0) and point B = (b , c)

∴ x1 = a , x2 = b and y1 = 0 and y2 = c

∴ The slope of AB is
`m=(c-0)/(b-a)=(c)/(b-a)`

# Slope of QR

∵ Point Q is (x1 , y1) and point R is (x2 , y2)

∵ Point Q = (2a , 0) and point R = (2b , 2c)

∴ x1 = 2a , x2 = 2b and y1 = 0 and y2 = 2c

∴ The slope of AB is
`m=(2c-0)/(2b-2a)=(2c)/(2(b-c))=(c)/(b-a)`

∵ The slopes of AB and QR are equal

∴ AB // QR

∵ AB is the line containing the midpoints of PQ and PR of Δ PQR

∵ QR is the third side of the triangle

∴ The line containing the midpoints of two sides of a triangle is parallel

to the third side