Two corners of an isosceles triangle are at (2 ,3 ) and (1 ,4 ). If the triangle's area is 64 , what are the lengths of the triangle's sides?

1 Answer
Oct 21, 2016

The 3 sides are 90.5, 90.5, and sqrt(2)

Explanation:

Let b = the length of the base from (2,3) to (1, 4)

b = sqrt((1 - 2)^2 + (4 - 3)^2)

b = sqrt(2)

This cannot be one of the equal sides, because the maximum area of such a triangle would occur, when it is equilateral, and specifically:

A = sqrt(3)/2

This conflicts with our given area, 64 units^2

We can use the Area to find the height of the triangle:

Area = (1/2)bh

64 = 1/2sqrt(2)h

h = 64sqrt(2)

The height forms a right triangle and bisects the base, therefore, we can use the Pythagorean theorem to find the hypotenuse:

c^2 = (sqrt(2)/2)^2 + (64sqrt(2))^2

c^2 = 8192.25

c~~ 90.5