### Video Transcript

๐ท๐ธ๐น is a triangle. ๐บ is a point on ๐ท๐น. Work out the area of triangle ๐ท๐ธ๐น. Give your answer to three significant figures.

So for this question, we can see that we have a big triangle ๐ท๐ธ๐น, but we also have
another line ๐ธ๐บ. And this line cuts our big triangle into two smaller triangles, ๐ท๐ธ๐บ and
๐ธ๐น๐บ. Now the question asks us to find the area of the big triangle ๐ท๐ธ๐น. But unfortunately, it doesnโt look like we have a lot of information about this
triangle. We have an angle of 70 degrees and an adjacent side length of 3.4 centimeters. Although it may look like we have two of the side lengths, when we look closely, this
measurement here of 2.5 centimeters corresponds to ๐บ๐น, not ๐ท๐น. So we donโt actually have two of the sides of our big triangle.

Now the question asks us to find the area of ๐ท๐ธ๐น. In this situation, the formula that we would usually turn to is the area of a
triangle is half times the base times the perpendicular height. Observing the diagram for triangle ๐ท๐ธ๐น, it doesnโt look like we have much luck
with this formula, since none of the angles appears to be a right angle, nor does it
look like weโll have an easy time finding the perpendicular height given the
information on the diagram.

I need to turn to another formula for the area of a triangle, half ๐๐ sin ๐ถ, where
๐ถ is one of the angles in the triangle and ๐ and ๐ are the two adjacent side
lengths of this angle. Now, weโre getting a little ahead of ourselves here. But itโs good to know what weโre aiming for.

This area formula will be the final stage in a multistep process. Again, we donโt have enough information about triangle ๐ท๐ธ๐น to implement this
formula yet. And so all of the preceding steps were to help us gather this information using what
weโve been given on the diagram.

Okay, first, a quick recap of some of the tools that will be helping us along the
way. The sine rule can be used in triangles where we have two angles and one of the angles
opposite these side lengths. Here, the sine rule can be used to find the angle opposite the other side length. It can also be used in triangles where we have two angles and one of the side lengths
opposite these angles. Here, the sine rule can be used to find the side length opposite the other angle.

The cosine rule can be used when we have an angle trapped between two side
lengths. In this situation, the cosine rule can be used to find the side length opposite the
angle we have. The cosine rule can also be used in triangles where all of the side lengths have been
given. In this situation, the cosine rule can be used to find any of the three angles in the
triangle.

Given these tools, now letโs walk through the method weโll be using to answer this
question. Step one, to begin, weโll be using the cosine rule on the smaller triangle
๐บ๐ธ๐น. Hopefully, here you can see we have a situation where we have a side, an angle, and a
side given. We can therefore use the cosine rule to find the missing side length, which on our
diagram is ๐ธ๐บ.

For step two, weโll continue on the same smaller triangle ๐ธ๐น๐บ. Using the newly discovered side length, we now have a situation where we can use the
sine rule and the side-side-angle case. This will allow us to find the missing angle marked here, which is angle ๐น๐ธ๐บ.

For step three, weโll be working on the big triangle ๐ท๐ธ๐น. One of the angles for the big triangle is given in the question. Here, weโll be in a position to find one of the other angles by adding the newly
discovered angle ๐น๐ธ๐บ to the other angle given by the question ๐ท๐ธ๐บ. By using the fact that angles in a triangle add up to 180 degrees, weโll be able to
find the third and final angle for our big triangle.

For step four, weโll be using the sine rule again for our big triangle ๐ท๐ธ๐น. This time, we have the other case of the sine rule, where we have an angle, an angle,
and a side. Given the newly discovered information, here weโll be able to use the sine rule to
find the side length opposite the other angle, which is side length ๐ท๐ธ. And here, finally, weโll be able to find the area of our big triangle ๐ท๐ธ๐น by using
the formula half ๐๐ sin ๐ถ that we mentioned earlier. This formula uses the same criteria as our first application of the cosine rule,
which is a known angle trapped between two known side lengths. At this stage in our calculations, we will indeed have this information about our big
triangle ๐ท๐ธ๐น. And we will therefore be able to use the formula to find the area of our
triangle.

Great! Now that we understand our method, letโs get to the calculations. Here we have the triangle ๐บ๐ธ๐น with the information that we currently know. And here we have the cosine rule in the form thatโs most useful for the
side-angle-side configuration. The notation used here is that a lowercase letter represents one of the side lengths
for our triangle, and the corresponding uppercase letter represents the angle
opposite that side length.

Here, we label our triangle to give some context. We have the side lengths ๐, ๐, and ๐ and the angles which are uppercase ๐ด, ๐ต,
and ๐ถ. We have chosen to label in this way because we know all of the information now on the
right-hand side of this equation: the side lengths lowercase ๐ and ๐ and the angle
uppercase ๐ด. This will allow us to substitute in the known values and solve to find the unknown
side length ๐, which is our desired side length on the triangle ๐ธ๐บ.

To make things extra clear, for this first calculation, here weโve the substitutions
weโll be making. Once we perform the substitutions, our cosine rule looks like this. For this type of question, make sure your calculator or device is set to degrees
instead of radians to avoid mistakes.

Now here for our next step of working, youโll notice that the value of cos 70 isnโt a
nice round number. So somewhere later in our calculations, weโll be approximating. Moving to the next step, we find that the value for ๐ธ๐บ squared is equal to 11.9956
dot dot dot. To find the value for ๐ธ๐บ, we take the square root of both sides of this
equation. And of course we ignore any negative solutions to the square root since ๐ธ๐บ is a
length.

Now here itโs worth noting, for questions like this, youโre usually given a little
bit of room for accuracy on your answer. But try to keep as many decimal places as you can on your calculator, especially
early on in your calculations, so you donโt lose too much to rounding.

And so performing a square root on our long decimal number, we find the value for
๐ธ๐บ is equal to 3.4634 dot dot dot centimeters. Great! Weโve now completed step one, and weโll keep this value to one side. Now moving on to step two, weโll be using the sine rule again in triangle ๐ธ๐น๐บ to
find the angle ๐น๐ธ๐บ, which is marked as a capital ๐ต on our diagram. And here in green we have the sine rule in the form thatโs most useful for the
side-side-angle criteria which weโll be using.

When applying the sine rule in this way, we look for a known side and angle pair,
which we have in ๐ and capital ๐ด. We also have another known side, which is ๐, and an angle which weโre interested in,
which is capital ๐ต. Although you can apply the sine rule to any of the side angle pairs, here we wonโt be
interested in the ๐ถs, and so weโll get rid of this part of the formula.

Now as we did before, weโll sub in the three known values capital ๐ด, lowercase ๐,
and lowercase ๐. Weโll then work through the equation to solve and find uppercase ๐ต, which is the
desired angle ๐น๐ธ๐บ. Substituting in, we find that sin of 70 degrees divided by 3.4634 is equal to sin of
angle ๐น๐ธ๐บ divided by 2.5.

To simplify, we first multiply both sides of the equation by 2.5. And we find that 0.6782 et cetera is equal to sin of angle ๐น๐ธ๐บ. To continue, we then take arc sine of both sides of the equation. And we find that angle ๐น๐ธ๐บ is equal to 42.71 et cetera degrees. With stage two complete, weโll keep this value to one side and move on to stage
three.

Weโre now looking at the sum of angles in a triangle for the big triangle ๐ท๐ธ๐น. So here we know that the angle marked ๐น๐ท๐ธ added to angle ๐ท๐ธ๐น and ๐ธ๐น๐ท. Looking at the diagram, we can easily see that angle ๐ท๐ธ๐น is comprised of two
smaller angles, one of which is the 30 degrees given by the question and the other
is angle ๐น๐ธ๐บ that we have just found, which is 42.71 et cetera degrees. We also know angle ๐ธ๐น๐ท as it is given as 70 degrees in the diagram.

Letโs sub these into the formula. So now that weโve substituted these values in, we want to solve for angle ๐น๐ท๐ธ. And we can do so by subtracting these two values from both sides of the equation. And performing this calculation, we find that angle ๐น๐ท๐ธ is equal to 37.28 dot dot
dot degrees.

Weโre now ready to move on to stage four of our method. Here we have our big triangle ๐ท๐ธ๐น drawn, with the relevant information
labeled. And here we have a different form of the sine rule, which is relevant to the
angle-angle-side situation that weโll be working with. Youโll know that weโre directly ignoring ๐ถ this time as itโs not relevant to our
calculations.

And here weโve labeled our triangle with lowercase letters for the side lengths and
uppercase letters for the angles. Weโve added a little dash to each of these symbols so as not to get confused with our
earlier triangle labeling with ๐s, ๐s, and ๐s. And to keep things consistent, weโll add this onto our sine rule as well.

So here we should be able to see that weโre aiming for side ๐ท๐ธ, which is
represented by ๐ prime. And the other three terms are known, since we know angle ๐ด, angle ๐ต, and side
length ๐. As weโve before, let sub everything in. We have ๐ท๐ธ divided by sin of 70 degrees is equal to 3.4 divided by sin of 37.28 dot
dot dot degrees. By multiplying both sides of the equation by sin of 70 degrees, weโre left with ๐ท๐ธ
on its own on the left-hand side.

We then work through our calculations, and we find that ๐ท๐ธ is equal to 5.2747
centimeters. With this piece of the puzzle, weโve completed step four, and weโre finally ready to
work out the area of our big triangle.

For step five, weโll be using the area of a triangle formula that we mentioned
earlier. Weโll also have the dashes to represent ๐ prime, ๐ prime, and capital ๐ถ prime to
be consistent with the diagram on the right. And finally, weโll collect the relevant information.

We can observe that the angle at ๐ท๐ธ๐น is represented by ๐ถ prime. Based on the information weโve previously found, we can easily see that this angle
๐ท๐ธ๐น is the sum of 30 degrees and 42.71 degrees, which is 72.71 degrees. Labeling this on our small diagram on the right, we can see that the criterion of
side-angle-side has been fulfilled, since we know side lengths ๐ prime, ๐ prime,
and angle ๐ถ prime.

We perform one final set of substitutions with our known values. And working through the numbers, we find that the area of triangle ๐ท๐ธ๐น is equal to
8.5617 centimeters squared. And donโt forget to add this squared to your unit since weโre working with an area
now, not a length.

As a final step, weโll note that our question asked for the answer to three
significant figures. We can count our significant figures from left to right. Our answer of 8.561 will round down to 8.56, and this is our answer. The area of the triangle ๐ท๐ธ๐น, to three significant figures, is 8.56 centimeters
squared.