# An isosceles triangle has sides A, B, and C with sides B and C being equal in length. If side A goes from (7 ,3 ) to (8 ,9 ) and the triangle's area is 45 , what are the possible coordinates of the triangle's third corner?

Nov 16, 2016

$\left(\frac{1635}{74} , \frac{264}{74}\right)$ or $\left(- \frac{525}{74} , \frac{624}{74}\right)$

#### Explanation:

Let's find the length of side A:

$A = \sqrt{{\left(8 - 7\right)}^{2} + {\left(9 - 3\right)}^{2}}$

$A = \sqrt{37}$

$A r e a = 45 = \frac{1}{2} \sqrt{37} h$

$h = 90 \frac{\sqrt{37}}{37}$

The height must go through the point $\left(7.5 , 6\right)$

The slope of the line for the height is:

$m = \frac{7 - 8}{9 - 3} = - \frac{1}{6}$

The equation of the line for height is:

$y = - \frac{1}{6} \left(x - 7.5\right) + 6$

Using the distance formula:

$h = \sqrt{{\left(x - 7.5\right)}^{2} + {\left(y - 6\right)}^{2}}$

${90}^{2} / 37 = {\left(x - 7.5\right)}^{2} + {\left(x - 7.5\right)}^{2} / 36$

${90}^{2} / 37 = \frac{36 {\left(x - 7.5\right)}^{2}}{36} + {\left(x - 7.5\right)}^{2} / 36$

${90}^{2} / 37 = 37 {\left(x - 7.5\right)}^{2} / 36$

${90}^{2} / {37}^{2} = {\left(x - 7.5\right)}^{2} / 36$

x - 7.5 = +-6(90/37)

x = 1635/74 and x = -525/74

y = 264/74 and y = 624/74