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

1 Answer
Mar 2, 2018

34.297"units"^2

Explanation:

We have a triangle that looks like:

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Here, we have that the angle C=pi/2, b=16"units" and angle A=pi/12.

Since all the angles of a triangle add up to pi, angle B=(5pi)/12.

According to the sine rule:

sinA/a=sinB/b

We have to solve for a. Inputting:

(sin(pi/12))/a=(sin((5pi)/12))/16

a=(16*sin(pi/12))/(sin((5pi)/12))

a=4.287"units"

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 a as the height of the triangle and b the base, and use the formula 1/2bh to solve for the area.
  • Find c and use Heron's Formula to solve for the area.

Just for kicks, let's do both!

Use Method Number 1:

A=1/2bh

A=1/2*16*4.29

A=34.297"units"^2

Use Method Number Two:

We must find c first. To do this, too, we have two methods:

Again, for kicks, I'm doing both.

color(white)(oleoleole)Sub-method Number 1:

color(white)(oleoleole)We have the Pythagoras' Theorem a^2+b^2=c^2.

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

color(white)(oleoleole)16^2+4.29^2=c^2

color(white)(oleoleole)c^2=274.404

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

Now for:

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

color(white)(oleoleole)We have the cosine rule: c^2=a^2+b^2-2abcosC

color(white)(oleoleole)We know the stuff we stated above, and color(white)(oleoleole)that C=pi/2

color(white)(oleoleole)Since cos(pi/2)=0, the cosine rule color(white)(oleoleole)color(white)(oleoleole)reduces to the Pythagoras' Theorem, which we color(white)(oleoleole)solved above, so skip it.

Now we go back to using Heron's Formula:

A=sqrt(s(s-a)(s-b)(s-c)), where s=(a+b+c)/2

Here, a=4.287, b=16, c=16.565. So:

s=(4.287+16+16.565)/2

s=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=sqrt(1176.216)

A=34.297"units"^2

Two different methods, same answer!