We have a triangle that looks like:
Wikipedia
Here, we have that the angle C=pi/2C=π2, b=16"units"b=16units and angle A=pi/12A=π12.
Since all the angles of a triangle add up to piπ, angle B=(5pi)/12B=5π12.
According to the sine rule:
sinA/a=sinB/bsinAa=sinBb
We have to solve for aa. Inputting:
(sin(pi/12))/a=(sin((5pi)/12))/16sin(π12)a=sin(5π12)16
a=(16*sin(pi/12))/(sin((5pi)/12))a=16⋅sin(π12)sin(5π12)
a=4.287"units"a=4.287units
Now there exist possibilities all around. There are a multitude of ways to solve for the area. Let's look at the two main ways of action.
- Take aa as the height of the triangle and bb the base, and use the formula 1/2bh12bh to solve for the area.
- Find cc and use Heron's Formula to solve for the area.
Just for kicks, let's do both!
Use Method Number 1:
A=1/2bhA=12bh
A=1/2*16*4.29A=12⋅16⋅4.29
A=34.297"units"^2A=34.297units2
Use Method Number Two:
We must find cc first. To do this, too, we have two methods:
Again, for kicks, I'm doing both.
color(white)(oleoleole)o≤o≤o≤Sub-method Number 1:
color(white)(oleoleole)o≤o≤o≤We have the Pythagoras' Theorem a^2+b^2=c^2a2+b2=c2.
color(white)(oleoleole)o≤o≤o≤We also know that a=4.29, b=16a=4.29,b=16. Inputting:
color(white)(oleoleole)o≤o≤o≤16^2+4.29^2=c^2162+4.292=c2
color(white)(oleoleole)o≤o≤o≤c^2=274.404c2=274.404
color(white)(oleoleole)o≤o≤o≤c=16.565"units"c=16.565units
Now for:
color(white)(oleoleole)o≤o≤o≤Sub-Method Number 2
color(white)(oleoleole)o≤o≤o≤We have the cosine rule: c^2=a^2+b^2-2abcosCc2=a2+b2−2abcosC
color(white)(oleoleole)o≤o≤o≤We know the stuff we stated above, and color(white)(oleoleole)o≤o≤o≤that C=pi/2C=π2
color(white)(oleoleole)o≤o≤o≤Since cos(pi/2)=0cos(π2)=0, the cosine rule color(white)(oleoleole)o≤o≤o≤color(white)(oleoleole)o≤o≤o≤reduces to the Pythagoras' Theorem, which we color(white)(oleoleole)o≤o≤o≤solved above, so skip it.
Now we go back to using Heron's Formula:
A=sqrt(s(s-a)(s-b)(s-c))A=√s(s−a)(s−b)(s−c), where s=(a+b+c)/2s=a+b+c2
Here, a=4.287, b=16, c=16.565a=4.287,b=16,c=16.565. So:
s=(4.287+16+16.565)/2s=4.287+16+16.5652
s=18.426s=18.426. Inputting all of that into Heron's Formula:
A=sqrt(18.426(18.426-4.287)(18.426-16)(18.426-16.565))A=√18.426(18.426−4.287)(18.426−16)(18.426−16.565)
A=sqrt(1176.216)A=√1176.216
A=34.297"units"^2A=34.297units2
Two different methods, same answer!