# An isosceles triangle has sides A, B, and C with sides B and C being equal in length. If side A goes from (1 ,4 ) to (5 ,8 ) and the triangle's area is 27 , what are the possible coordinates of the triangle's third corner?

May 29, 2018

(9.75, -0.75) or (-3.75, 12.75)

#### Explanation:

First picture side A, common to two possible isosceles triangles, from (1,4) to (5,8). By Pythagoras we see that this side has length $4 \sqrt{2}$.

Now draw the line perpendicular to side A which also bisects side A, up until it reaches where sides B and C of the isosceles triangle meet; this new line is the height of the triangle.

Now you will realise that the area of the triangle, 27, is equal to $\frac{1}{2} b h = \frac{1}{2} \cdot 4 \sqrt{2} \cdot h$. Solving for $h$, we see that $h = \frac{27 \sqrt{2}}{4}$.

The final thing to note is that since this line is perpendicular to side A, which has a gradient of 1, this line has a gradient of -1. Hence, the horizontal and vertical components of the triangle connecting the midpoint of A and the other point of the isosceles triangle are equal in magnitude. Hence we can set up the equation $\sqrt{2 {x}^{2}} = \frac{27 \sqrt{2}}{4}$, which when solved gives $x = \pm \frac{27}{4.}$

Now all that is left is to add this value of $x$ to the midpoint $\left(3 , 6\right)$ of side A, taking care with the signs, to obtain the possible coordinates of the final point.

The point of the possible triangle to the bottom right of the midpoint is $\left(3 + \frac{27}{4} , 6 - \frac{27}{4}\right) = \left(9.75 , - 0.75\right)$ and the point to the top left of the midpoint is $\left(3 - \frac{27}{4} , 6 + \frac{27}{4}\right) = \left(- 3.75 , 12.75\right)$, both as required.