Answer: Either 16 cm or 30 cm.
Work Shown
Consider a triangle with sides a,b,c
- a = x = unknown
- b = 17
- c = 17
s = semi-perimeter = half of the perimeter
s = (a+b+c)/2
s = (x+17+17)/2
s = (x+34)/2
s = 0.5x+17
Now use Heron's Formula
A = sqrt(s*(s-a)*(s-b)*(s-c))
120 = sqrt((0.5x+17)*((0.5x+17)-x)*((0.5x+17)-17)*((0.5x+17)-17))
120 = sqrt((0.5x+17)*(-0.5x+17)*(0.5x)*(0.5x))
I'll skip a few steps, but the four solutions to that equation are
x = -30, x = -16, x = 16, x = 30
Ignore the negative x values. A negative side length doesn't make sense.
The only possible side lengths for the base are 16 cm and 30 cm.
We could have an isosceles triangle with sides 16 cm, 17 cm, 17 cm.
OR
We could have an isosceles triangle with sides 17 cm, 17 cm,30 cm.
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Another approach
x = half of the unknown base
h = height
Draw an isosceles triangle that points upward. Split it in two along a vertical line through the top most point. This creates two smaller triangles that are mirror clones of each other (in other words they are congruent). Furthermore, they are right triangles.
Focus on one of those smaller right triangles.
base = x
height = h
area = 0.5*base*height = 0.5xh
The two smaller triangles combine to form a total area of 2*0.5*xh = xh
Since the area is 120, we know that xh = 120 and h = 120/x.
Now use the pythagorean theorem to say:
a^2+b^2 = c^2
x^2+h^2 = 17^2
x^2 + (120/x)^2 = 17^2
I'll skip steps a bit. After isolating x, you should have these four solutions
x = -15, x = -8, x = 8, x = 15
Toss out the negative x values.
x = 8 is half the base, so the full base would be 2x = 2*8 = 16
or
x = 15 is half the base, so the full base would be 2x = 2*15 = 30
We arrive at the same solutions as earlier.