### Video Transcript

Find the length of side 𝑎 of the
triangle shown.

In our triangle, we see this side
length opposite an angle of 103 degrees. Along with this, based on these
markings on the other two sides of the triangle, we can tell that they have the same
length. So this side length here is 7.1
centimeters also. As a side note, this means we’re
working with a special kind of triangle called an isosceles triangle. But anyway, our mission is to solve
for this side length here, and we can do it using a rule called the cosine rule,
also sometimes called the law of cosines.

This rule tells us that given a
triangle with interior angles marked out capital 𝐴, 𝐵, and 𝐶 and corresponding
side lengths lowercase 𝑎 and 𝑏 and 𝑐, we can solve for the square of one of the
sides by adding together the square of the other two sides and then subtracting from
that two times those other two sides multiplied by the cos of an angle we’ve called
capital 𝐴, where in our triangle, this angle capital 𝐴 is opposite the side length
lowercase 𝑎 that we’re solving for. So in our actual triangle over
here, to solve for this side length, we’ll apply the cosine rule.

To do this, we’ll need to identify
what the other two side lengths, they’re called 𝑏 and 𝑐 in this equation, and the
angle opposite the side length we want to solve for, called capital 𝐴 here,
are. Looking at our triangle, we see
that angle opposite the side length we’re solving for is 103 degrees. So that’s our angle capital 𝐴. And then, as far as identifying 𝑏
and 𝑐, these values in our cosine rule equation, we see that the other two side
lengths in our triangle are both the same. In this case then, both 𝑏 and 𝑐
are 7.1 centimeters. This is the case because, as we saw
earlier, we’re working with an isosceles triangle.

Knowing all this, we can now plug
in these values into their place on the right-hand side of our cosine rule
expression. And once we’ve done this, we’re
very close to being able to solve for the unknown side length 𝑎. The one thing we don’t want to
forget, though, is that right now we have an expression for 𝑎 squared. To solve just for 𝑎, we’ll want to
take the square root of both sides, where on the left, this square root and the
square term will cancel one another out. And now, when we enter this
expression on the right-hand side into a calculator, rounding to two significant
figures, we find a result of 11 centimeters. This is the length of side 𝑎 in
our triangle.