Two corners of an isosceles triangle are at (8 ,5 ) and (9 ,1 ). If the triangle's area is 12 , what are the lengths of the triangle's sides?

Jul 23, 2018

color(maroon)("Lengths of the triangle "a = sqrt 17, b = sqrt(2593 / 68), c = sqrt(2593 / 68)

Explanation:

color(red)( B(8,5), C(9,1), A_t = 12

let $\overline{A D} = h$

$\overline{B C} = a = \sqrt{{\left(9 - 8\right)}^{2} + {\left(1 - 5\right)}^{2}} = \sqrt{17}$

Area of triangle " A_t = 12 = (1/2) a *h = (sqrt17 h)/2

$h = \frac{24}{\sqrt{17}}$

$\overline{A C} = \overline{A B} = b = \sqrt{{\left(\frac{a}{2}\right)}^{2} + {h}^{2}}$

$b = \sqrt{{\left(\frac{\sqrt{17}}{2}\right)}^{2} + {\left(\frac{24}{\sqrt{17}}\right)}^{2}}$

$b = \sqrt{\frac{17}{4} + \frac{576}{17}} = \sqrt{\frac{2593}{68}}$