Raymond B. answered • 03/30/20
Math, microeconomics or criminal justice
This is an annoying problem, as rounding errors or just not enough decimal places can throw off the answer.
But the following seems to be nearly perfect, when checked using Heron's formula
a=12.452
b=7.212
c=11.203
With heron's formula, the square root of the semi-perimeter times each side subtracted from that semi-perimeter is almost exactly 40 for the area. This has to be the correct answer If you think you have another answer that's correct, check it with Heron's formula and if you aren't close to 40, then you must be wrong.
It's almost a 30-60-90 right triangle with sides in a ratio of 1:1.73:2
shortest sides multiply to twice the Area = 2(40)=80
n x 1.73n = 80
n squared = 80/1.732
n = 4sqr5/4th root of 3 = about 4(2.24)/1.316 = 8.96/1.316 = 6.81,
so sides are 6.6, 6.6 x sqr3 and 13.2
But for more precision
use side a as the base and let h=height, then ah = 2x area = 80
h=80/a
put the triangle on side a as the base and drop a vertical from angle A to the base.
It divides the base in two parts, x and a-x
tanB=h/x tan32 = h/x = 0.625 = 5/8
tanC=h/(a-x) = tan63= 1.96
h/x = .625
h/a-x)=1.96
h=80/a
3 equations 3 unknowns,
h= .625x = 1.96(a-x)
h= 1.96a -1.96x = 80/a
1.96a2 -1.96ax = 80
h=80/a = .625x
80 =.625ax
ax = 80/(5/8)= 640/5=128
1.96a2 -1.96(128) = 80
1.96a2 = 80 + 250.88 = 330.88
a2 = 330.88/1.96 = 168.82
a= square root of 168.82 = 13
b= a/2 =6.5
c= 6.5 x 1.732 = 11,26
which is very close to the approximation when we assumed a right triangle.
h=80/a= 80/13
Area = 40 = 1/2 x 80/13 x 13
But when you plug these answers into heron's formula, it doesn't get quite close enough to 40 for the area.
Due to rounding errors and not enough decimal places, the above answers are a little off. Best answers that work with Heron's formula are:
a=12.452
b=7.212
c=11.203
You can try rounding that to
a=12.5
b=7.2
c=11.2
but they don't work quite so well in Heron's formula
