Question

Find the equations of the medians of a triangle, the coordinates of whose vertices are (−1, 6), (−3, −9) and (5, −8).

Answer 1

Let A (−1, 6), B (−3, −9), and C (5, −8) be the coordinates of the given triangle.

Let D, E and F be midpoints of BC, CA and AB, respectively.

So, the coordinates of D, E, and F are:

D = \[\left( \frac{- 3 + 5}{2}, \frac{- 9 - 8}{2} \right)\]

∴ D = \[\left( 1, \frac{- 17}{2} \right)\]

E = \[\left( \frac{- 1 + 5}{2}, \frac{6 - 8}{2} \right)\]

∴ E = (2, −1)

F = \[\left( \frac{- 1 - 3}{2}, \frac{6 - 9}{2} \right)\]

∴ F = \[\left( - 2, - \frac{3}{2} \right)\]

Median AD passes through,

A(−1, 6) and D\[\left( 1, - \frac{17}{2} \right)\]

So, its equation is:

\[y - 6 = \frac{- \frac{17}{2} - 6}{1 + 1}\left( x + 1 \right)\]

⇒ 4y − 24 = −29x − 29

⇒ 29x + 4y + 5 = 0

Median BE passes through B (−3, −9) and E(2, −1)

So, its equation is:

\[y + 9 = \frac{- 1 + 9}{2 + 3}\left( x + 3 \right)\]

⇒ 5y + 45 = 8x + 24

⇒ 8x − 5y − 21 = 0

Median CF passes through,

C (5, −8) and F \[\left( - 2, - \frac{3}{2} \right)\]

So, its equation is:

\[y + 8 = \frac{- \frac{3}{2} + 8}{- 2 - 5}\left( x - 5 \right)\]

⇒ −14y − 112 = 13x − 65

⇒13x + 14y + 47 = 0