Question

Two vertices of triangle are (–1, 4) and (5, 2) and
medians intersect at (0, –3), then the third vertex is
A (4, 15).
B (3, 15).
C (– 4,15).
D (– 4, – 15).​

Answer 1

Given:

• Vertices of triangle =(-1,4) and (5,2)

• Meridian intersect =(0,-3)

________________________________________

To Find :

• Third vertex of triangle =?

________________________________________

Solution :

>> Given vertices (-1,4) and (5,2) ,the meridian meets at centroid =(0,-3)

>> We know that in a triangle with vertices $\rm{(x_{1},y_{1}),(x_{2},y_{2}),(x_{3},y_{3}) }$ then the centroid will be $\rm{\bigg(\dfrac{x_{1}+x_{2}+x_{3}}{3} , \dfrac{y_{1}+y_{2}+y_{3}}{3} \bigg) }$.

>> Let the third vertex be (a, b) .

We have,

$\sf{(0,-3)=\bigg(\dfrac{-1+5+a}{3} ,\dfrac{4+2+b}{3}\bigg) }$

$\implies\sf{0=\dfrac{-1+5+a}{3}}$

$\implies \sf \: 0 = \frac{4 + a}{ 3} \\ \\ \implies \sf \: a + 4 = 0 \\ \\ \implies \boxed{ \sf{a = - 4}} \\ \\ \\ \implies \sf \: - 3 = \frac{4 + 2 + b}{3} \\ \\ \implies \sf \: \: - 3 = \frac{6 + b}{3} \\ \\ \implies \sf \: - 9 = 6 + b \\ \\ \implies \sf \: - 9 - 6 - b = 0 \\ \\ \implies \boxed{ \sf{b = - 15}}$

Therefore,

(-4,-15) are the third vertex of a triangle.

________________________________________

Additional information :

• If the 3 points are given the question tell us to find centroid of the 3 points we use centroid formula :

Centroid Formula:

$\red\bigstar\boxed{\sf{(x, y) =\dfrac{\bigg(x_{1}+x_{2}+x_{3}}{3} ,\dfrac{y_{1}+y_{2}+y_{3}\bigg)}{3}}}$

• If Two point are given the question ask to find midpoint of the line we use midpoint formula :

Midpoint formula :

$\purple\bigstar\boxed{\sf{(x, y) =\bigg(\dfrac{x_{1}+x_{2}}{2} ,\dfrac{y_{1}+y_{2}}{2}\bigg) }}$

• If the question ask to find distance between two points we use distance formula :

Distance Formula :

$\blue\bigstar\boxed{\sf{AB=\sqrt{(y_{2}-y_{1})^2+(x_{2}-x_{1})^2}}}$

• if the question ask to find the ratio of m:n in which it internally divides between points we use section formula :

Section Formula :

$\green\bigstar\boxed{\sf{(x, y) =\bigg(\dfrac{mx_{2}+nx_{1}}{m+n} , \dfrac{my_{2}+ny_{1}}{m+n}\bigg) }}$