𝐷𝐸𝐹 is a triangle. 𝐺 is a point on 𝐷𝐹. Work out the area of triangle 𝐷𝐸𝐹. Give your answer to three significant figures.
So for this question, we can see that we have a big triangle 𝐷𝐸𝐹, but we also have
another line 𝐸𝐺. And this line cuts our big triangle into two smaller triangles, 𝐷𝐸𝐺 and
𝐸𝐹𝐺. Now the question asks us to find the area of the big triangle 𝐷𝐸𝐹. But unfortunately, it doesn’t 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 𝐺𝐹, not 𝐷𝐹. So we don’t actually have two of the sides of our big triangle.
Now the question asks us to find the area of 𝐷𝐸𝐹. 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 𝐷𝐸𝐹, it doesn’t 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 we’ll 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 𝑎𝑏 sin 𝐶, where
𝐶 is one of the angles in the triangle and 𝑎 and 𝑏 are the two adjacent side
lengths of this angle. Now, we’re getting a little ahead of ourselves here. But it’s good to know what we’re aiming for.
This area formula will be the final stage in a multistep process. Again, we don’t have enough information about triangle 𝐷𝐸𝐹 to implement this
formula yet. And so all of the preceding steps were to help us gather this information using what
we’ve 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
Given these tools, now let’s walk through the method we’ll be using to answer this
question. Step one, to begin, we’ll be using the cosine rule on the smaller triangle
𝐺𝐸𝐹. 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 𝐸𝐺.
For step two, we’ll continue on the same smaller triangle 𝐸𝐹𝐺. 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 𝐹𝐸𝐺.
For step three, we’ll be working on the big triangle 𝐷𝐸𝐹. One of the angles for the big triangle is given in the question. Here, we’ll be in a position to find one of the other angles by adding the newly
discovered angle 𝐹𝐸𝐺 to the other angle given by the question 𝐷𝐸𝐺. By using the fact that angles in a triangle add up to 180 degrees, we’ll be able to
find the third and final angle for our big triangle.
For step four, we’ll be using the sine rule again for our big triangle 𝐷𝐸𝐹. 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 we’ll be able to use the sine rule to
find the side length opposite the other angle, which is side length 𝐷𝐸. And here, finally, we’ll be able to find the area of our big triangle 𝐷𝐸𝐹 by using
the formula half 𝑎𝑏 sin 𝐶 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 𝐷𝐸𝐹. And we will therefore be able to use the formula to find the area of our
Great! Now that we understand our method, let’s get to the calculations. Here we have the triangle 𝐺𝐸𝐹 with the information that we currently know. And here we have the cosine rule in the form that’s 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 𝑎, 𝑏, and 𝑐 and the angles which are uppercase 𝐴, 𝐵,
and 𝐶. 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 𝑏 and 𝑐 and the angle
uppercase 𝐴. This will allow us to substitute in the known values and solve to find the unknown
side length 𝑎, which is our desired side length on the triangle 𝐸𝐺.
To make things extra clear, for this first calculation, here we’ve the substitutions
we’ll 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, you’ll notice that the value of cos 70 isn’t a
nice round number. So somewhere later in our calculations, we’ll be approximating. Moving to the next step, we find that the value for 𝐸𝐺 squared is equal to 11.9956
dot dot dot. To find the value for 𝐸𝐺, we take the square root of both sides of this
equation. And of course we ignore any negative solutions to the square root since 𝐸𝐺 is a
Now here it’s worth noting, for questions like this, you’re 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 don’t lose too much to rounding.
And so performing a square root on our long decimal number, we find the value for
𝐸𝐺 is equal to 3.4634 dot dot dot centimeters. Great! We’ve now completed step one, and we’ll keep this value to one side. Now moving on to step two, we’ll be using the sine rule again in triangle 𝐸𝐹𝐺 to
find the angle 𝐹𝐸𝐺, which is marked as a capital 𝐵 on our diagram. And here in green we have the sine rule in the form that’s most useful for the
side-side-angle criteria which we’ll be using.
When applying the sine rule in this way, we look for a known side and angle pair,
which we have in 𝑎 and capital 𝐴. We also have another known side, which is 𝑏, and an angle which we’re interested in,
which is capital 𝐵. Although you can apply the sine rule to any of the side angle pairs, here we won’t be
interested in the 𝐶s, and so we’ll get rid of this part of the formula.
Now as we did before, we’ll sub in the three known values capital 𝐴, lowercase 𝑎,
and lowercase 𝑏. We’ll then work through the equation to solve and find uppercase 𝐵, which is the
desired angle 𝐹𝐸𝐺. Substituting in, we find that sin of 70 degrees divided by 3.4634 is equal to sin of
angle 𝐹𝐸𝐺 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 𝐹𝐸𝐺. To continue, we then take arc sine of both sides of the equation. And we find that angle 𝐹𝐸𝐺 is equal to 42.71 et cetera degrees. With stage two complete, we’ll keep this value to one side and move on to stage
We’re now looking at the sum of angles in a triangle for the big triangle 𝐷𝐸𝐹. So here we know that the angle marked 𝐹𝐷𝐸 added to angle 𝐷𝐸𝐹 and 𝐸𝐹𝐷. Looking at the diagram, we can easily see that angle 𝐷𝐸𝐹 is comprised of two
smaller angles, one of which is the 30 degrees given by the question and the other
is angle 𝐹𝐸𝐺 that we have just found, which is 42.71 et cetera degrees. We also know angle 𝐸𝐹𝐷 as it is given as 70 degrees in the diagram.
Let’s sub these into the formula. So now that we’ve substituted these values in, we want to solve for angle 𝐹𝐷𝐸. And we can do so by subtracting these two values from both sides of the equation. And performing this calculation, we find that angle 𝐹𝐷𝐸 is equal to 37.28 dot dot
We’re now ready to move on to stage four of our method. Here we have our big triangle 𝐷𝐸𝐹 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 we’ll be working with. You’ll know that we’re directly ignoring 𝐶 this time as it’s not relevant to our
And here we’ve labeled our triangle with lowercase letters for the side lengths and
uppercase letters for the angles. We’ve added a little dash to each of these symbols so as not to get confused with our
earlier triangle labeling with 𝑎s, 𝑏s, and 𝑐s. And to keep things consistent, we’ll add this onto our sine rule as well.
So here we should be able to see that we’re aiming for side 𝐷𝐸, which is
represented by 𝑎 prime. And the other three terms are known, since we know angle 𝐴, angle 𝐵, and side
length 𝑏. As we’ve before, let sub everything in. We have 𝐷𝐸 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, we’re left with 𝐷𝐸
on its own on the left-hand side.
We then work through our calculations, and we find that 𝐷𝐸 is equal to 5.2747
centimeters. With this piece of the puzzle, we’ve completed step four, and we’re finally ready to
work out the area of our big triangle.
For step five, we’ll be using the area of a triangle formula that we mentioned
earlier. We’ll also have the dashes to represent 𝑎 prime, 𝑏 prime, and capital 𝐶 prime to
be consistent with the diagram on the right. And finally, we’ll collect the relevant information.
We can observe that the angle at 𝐷𝐸𝐹 is represented by 𝐶 prime. Based on the information we’ve previously found, we can easily see that this angle
𝐷𝐸𝐹 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 𝑎 prime, 𝑏 prime,
and angle 𝐶 prime.
We perform one final set of substitutions with our known values. And working through the numbers, we find that the area of triangle 𝐷𝐸𝐹 is equal to
8.5617 centimeters squared. And don’t forget to add this squared to your unit since we’re working with an area
now, not a length.
As a final step, we’ll 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 𝐷𝐸𝐹, to three significant figures, is 8.56 centimeters