# An isosceles triangle has sides A, B, and C with sides B and C being equal in length. If side A goes from (4 ,9 ) to (1 ,0 ) and the triangle's area is 32 , what are the possible coordinates of the triangle's third corner?

Jan 8, 2018

Coordinates of $A = \left(0.6642 , 2.3667\right)$

#### Explanation:

$a = \sqrt{{\left(4 - 1\right)}^{2} + {\left(9 - 0\right)}^{2}} = 9.4868$

${A}_{t} = \left(\frac{1}{2}\right) a h$

$h = \frac{2 \cdot 32}{9.4868} = 6.7462$

$\tan \theta = m = \frac{h}{\frac{a}{2}} = \frac{2 \cdot 6.7462}{9.4868} = 1.4222$

Equation of line passing through B and slope m is

$y - 9 = 1.4222 \left(x - 4\right)$

$y - 1.4222 x = 3.3112$ Eqn (1)

Equation of line passing through C and slope (-m) is

$y - 0 = - 1.4222 \left(x - 1\right)$

$y + 1.4222 x = 1.4222$ Eqn (2)

Solving Eqns (1), (2) we get the coordinates of point A.

$x = 0.6642 , y = 2.3667$