Two corners of an isosceles triangle are at #(7 ,5 )# and #(3 ,6 )#. If the triangle's area is #6 #, what are the lengths of the triangle's sides?
1 Answer
There's a couple ways to do it; the way with the fewest steps is explained below.
The question is ambiguous about which two sides are the same length. In this explanation, we will assume the two sides of equal length are the ones yet to be found.
Explanation:
One side length we can figure out just from the coordinates we've been given.
Then we can use the formula for area of a triangle in terms of its side lengths to figure out
where
Since
Substituting this into the area formula above, as well as
Our solution is
Footnote 1:
It is possible to have a triangle with two sides of length
Footnote 2:
We could also have solved this question by finding the coordinates of the 3rd point. This would have involved:
a) finding the length of the known side
b) finding the slope
c) finding the midpoint
d) finding the "height"
e) finding the slope of the height using
f) using both the slope-point formula
g) after combining these two equations, simplifying yields
h) plugging in the known values for
i) using one of the two equations in (f) to find
j) using the distance formula to find the remaining (identical) side lengths
You can see why the first method is easier.