Question

The midpoints of the sides BC, CA and AB of a $△ABC$ are D(3, 4), E(8, 9) and F(6, 7) respectively. Find the coordinates of hte vertices of the triangle.

Answer 1

A(a,b) B(c,d) C(e,f)

Midpoint of AB:
$(6,7)=\left(\frac{a+c}{2}\right),\left(\frac{b+d}{2}\right)$
a+c=12........(1) and b+d=14.......(2)

Midpoint of BC:
$(3,4)=\left(\frac{c+e}{2}\right),\left(\frac{d+f}{2}\right)$
c+e=6........(3) and d+f=8.........(4)

Midpoint of AC:
$(8,9)=\left(\frac{a+e}{2}\right),\left(\frac{b+f}{2}\right)$
a+e=16.........(5) and b+f=18........(6)

(1)-(3)=>
a+c-c-e=12-6
a-e=6........(7)

(5)+(7):
a+e+a-e=16+6
2a=22
a=11
substitute a=11 in (1)
c=1
substitute a=11 in (5)
e=5

(2)-(4)=>
b+d-d-f=14-8
b-f=6..........(8)
(6)+(8):
b+f+b-f=18+6
2b=24
b=12
substitute b=12 in (2)
d=2
substitute d=2 in (4)
f=4

A(11,12) B(1,2) C(5,4)