# An isosceles triangle has sides A, B, and C with sides B and C being equal in length. If side A goes from (4 ,3 ) to (8 ,9 ) and the triangle's area is 36 , what are the possible coordinates of the triangle's third corner?

Jul 8, 2017

The coordinates of the third corner are $\left(14.3 , 0.5\right)$ or $\left(- 2.3 , 11.5\right)$

#### Explanation:

Let the coordinates of the third corner be $= \left(x , y\right)$

The mid point of side $A$ is $= \left(\frac{8 + 4}{2} , \frac{9 + 3}{2}\right) = \left(6 , 6\right)$

We apply Pythagoras' theorem to the triangles

${\left(x - 4\right)}^{2} + {\left(y - 3\right)}^{2} = {\left(6 - 4\right)}^{2} + {\left(6 - 3\right)}^{2} + {\left(x - 6\right)}^{2} + {\left(y - 6\right)}^{2}$

${x}^{2} - 8 x + 16 + {y}^{2} - 6 y + 9 = 4 + 9 + {x}^{2} - 12 x + 36 + {y}^{2} - 12 y + 36$

$- 8 x - 6 y + 25 = 85 - 12 x - 12 y$

$4 x + 6 y = 60$

$2 x + 3 y = 30$...........................$\left(1\right)$

We now write the second equation

The area is

$a = \frac{1}{2} \cdot b \cdot h$

$\frac{1}{2} \left(\sqrt{{\left(8 - 4\right)}^{2} + {\left(9 - 3\right)}^{2}}\right) \left(\sqrt{{\left(x - 6\right)}^{2} + {\left(y - 6\right)}^{2}}\right) = 36$

$\sqrt{52} \cdot \left(\sqrt{{\left(x - 6\right)}^{2} + {\left(y - 6\right)}^{2}}\right) = 72$

$52 \left({\left(x - 6\right)}^{2} + {\left(y - 6\right)}^{2}\right) = {72}^{2} = 5184$

${\left(x - 6\right)}^{2} + {\left(y - 6\right)}^{2} = 99.7$.............................$\left(2\right)$

Solving for $x$ and $y$ in equations $\left(1\right)$ and $\left(2\right)$

${\left(x - 6\right)}^{2} + {\left(\frac{30 - 2 x}{3} - 6\right)}^{2} = 99.7$

${x}^{2} - 12 x + 36 + \frac{144 - 48 x + 4 {x}^{2}}{9} = 99.7$

$9 {x}^{2} - 108 x + 324 + 144 - 48 x + 4 {x}^{2} = 897.3$

$13 {x}^{2} - 156 x - 429.3 = 0$

$x = \frac{156 \pm \sqrt{{156}^{2} - 4 \cdot 13 \cdot \left(- 429.3\right)}}{26}$
${x}_{1} = \frac{156 + 216}{26} = 14.3$
${x}_{2} = \frac{156 - 216}{26} = - 2.3$
${y}_{1} = \frac{30 - 2 \cdot 14.3}{3} = 0.5$
${y}_{2} = \frac{30 + 2 \cdot 2.3}{3} = 11.5$