A triangle has sides A, B, and C. The angle between sides A and B is (pi)/2π2 and the angle between sides B and C is pi/12π12. If side B has a length of 16, what is the area of the triangle?

1 Answer
Mar 2, 2018

34.297"units"^234.297units2

Explanation:

We have a triangle that looks like:

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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=16sin(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=12164.29

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)oooSub-method Number 1:

color(white)(oleoleole)oooWe have the Pythagoras' Theorem a^2+b^2=c^2a2+b2=c2.

color(white)(oleoleole)oooWe also know that a=4.29, b=16a=4.29,b=16. Inputting:

color(white)(oleoleole)ooo16^2+4.29^2=c^2162+4.292=c2

color(white)(oleoleole)oooc^2=274.404c2=274.404

color(white)(oleoleole)oooc=16.565"units"c=16.565units

Now for:

color(white)(oleoleole)oooSub-Method Number 2

color(white)(oleoleole)oooWe have the cosine rule: c^2=a^2+b^2-2abcosCc2=a2+b22abcosC

color(white)(oleoleole)oooWe know the stuff we stated above, and color(white)(oleoleole)ooothat C=pi/2C=π2

color(white)(oleoleole)oooSince cos(pi/2)=0cos(π2)=0, the cosine rule color(white)(oleoleole)ooocolor(white)(oleoleole)oooreduces to the Pythagoras' Theorem, which we color(white)(oleoleole)ooosolved above, so skip it.

Now we go back to using Heron's Formula:

A=sqrt(s(s-a)(s-b)(s-c))A=s(sa)(sb)(sc), 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(18.4264.287)(18.42616)(18.42616.565)

A=sqrt(1176.216)A=1176.216

A=34.297"units"^2A=34.297units2

Two different methods, same answer!