# An isosceles triangle has sides A, B, and C with sides B and C being equal in length. If side A goes from (2 ,5 ) to (8 ,7 ) and the triangle's area is 16 , what are the possible coordinates of the triangle's third corner?

Apr 12, 2018

The coordinates of A is $A \left(0.4545 , 9.8182\right)$

#### Explanation:

Let the coordinates of A, B and C be $\left(x , y\right)$, $\left(2 , 5\right)$ and $\left(8 , 7\right)$ respectively.

Given that, $A B = A C$

$\rightarrow A {B}^{2} = A {C}^{2}$

$\rightarrow {\left(2 - x\right)}^{2} + {\left(5 - y\right)}^{2} = {\left(8 - x\right)}^{2} + {\left(7 - y\right)}^{2}$

$\rightarrow 4 - 4 x + \cancel{{x}^{2}} + 25 - 10 y + \cancel{{y}^{2}} = 64 - 16 x + \cancel{{x}^{2}} + 49 - 14 y \cancel{+ {y}^{2}}$

$\rightarrow 3 x + 2 y = 21$......[1]

Also, Area of $\Delta A B C = 16$

$\rightarrow \frac{1}{2} | \left(x , y , 1\right) , \left(2 , 5 , 1\right) , \left(8 , 7 , 1\right) | = 16$

$\rightarrow x \left(5 - 7\right) - y \left(2 - 8\right) + 1 \left(14 - 40\right) = 32$

$\rightarrow - 2 x + 6 y - 26 = 32$

$\rightarrow x - 3 y = 4$.....[2]

Solving [1] and [2], we get,

$x = 0.4545 \mathmr{and} y = 9.8182$

So, the coordinates of A is $\left(0.4545 , 9.8182\right)$.