Question

Find the third vertex of a triangle if its two vertices are
(-1, 4) and (5,2) and mid-point of one side is (0,3 ).

Answer 1

We know that $A(-1,4),B(0,3)$ which is

given $C(5,2)$

Let the third vertex $(x_1,y_1)$. There are two join case

we join vertex $(x_1,y_1)$ with $(-1,4)$ mid of this segment is $(0,3)$ then

Using mid value formula

$\frac{x_1+x_2}{2}=0,\frac{y_1+y_2}{2}=3$

$x_1+x_2=0,y_1+y_2=6$

$x_1=-x_2,y_1=6-y_2$

$x_1=1,y_1=6-4=2$

$(x_1,y_1)=(1,2)$

i.e third vertex $D(1,2)=(x_1,y_1)$

$E(x_1^{\prime },y_1^{\prime })$ join with $C(5,2)$ vertex and mid point $(0,3)$

Now we can write $D(0,3)$ is mid point of line segment between $(x_1^{\prime },y_1^{\prime })$ and $(5,2)$

$\frac{x_1^{\prime }+5}{2}=0,\frac{y_1^{\prime }+2}{2}=3$

$x_1^{\prime }=-5$, y_{1}'=4$

$(x_1^{\prime },y_1^{\prime })=(-5,4)$

another vertex $(-5,4)E$

The required vertex could be either $(5,4)$ and $(1,2)$