# 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?

##### 1 Answer

(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

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

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

Now all that is left is to add this value of

The point of the possible triangle to the bottom right of the midpoint is