Lesson video

Hi, I'm Mr. Bond, and in this lesson, we're going to use the area of a triangle and the formula for the area of a triangle.

The area is equal to 1/2 ab sin C to find the value of a missing length, a or b.

Let's take a look at our first example.

We're told that the area of the triangle is 52 centimetres squared and we need to work out the value of the length x.

Well, let's start with what we know.

We know that the area of a triangle is given by 1/2 ab sin C.

Now we can substitute in what we know and then solve to find x.

So we know that the area is 52.

We know that a is 10 centimetres.

We don't know b, this is X.

And we know that the angle C is 70 degrees.

So substituting into the formula gives this.

Now we need to solve for X.

The first thing that I can see we can do, is perform the part of the calculation.

1/2 multiplied by 10.

This is equal to five.

So now we have, 52, is equal to five multiplied by X, multiplied by sin 70.

Now dividing both sides of this equation by five, gives, 52/5 is equal to x multiplied by sin 70 and now dividing by sin 70, gives 52/5 sin 70.

is equal to X, and now if we input this calculation into our calculators, it will give the result, X is equal to 11.

1 centimetres to three significant figures.

Here's a question for you to try.

Pause the video to complete your task and resume the video once you've finished.

Here's the solution.

This was just like the previous example.

Use the area formula, substituting what you know and then use it to find the missing value AC which you might've called a variable X, for example Here's another question for you to try.

Pause the video to complete your task and resume the video once you finished.

Here's the answer.

The first key piece of information is that the area of the triangle is twice the area of the rectangle.

So we have to start by finding the area of the rectangle multiplying this by two, and then this is the area of the triangle.

Here we have a triangle with side lengths, six centimetres and 2.

5 centimetres and an included angle of 82 degrees.

I want to know which length is a and which is b.

Pause the video to have a think and resume the video when you've finished What did you think? Well, you could have said that a was equal to six centimetres and b was equal to 2.

5 centimetres, or there was another option.

You could also have said that a is equal to 2.

5 centimetres and b is equal to six centimetres.

The important part is that we choose a and b in the area formula as being the two sides around the included angle that we use.

Here's another example.

We're told that the area of the triangle is equal to the area of the square, and we're asked to work out the length of the edge at which they're joined.

So, what can we work out? Well, we know that the area of the triangle will be given by 1/2 ab sin C.

So, substituting in those values will give that the area of the triangle is equal to 1/2 multiplied by eight multiplied by K multiplied by sin 38.

And we can also say that the area of the square is given by K squared because K, is of course the length at which they're joined.

We're told that these two areas are equal.

So let's put those two expressions equal to each other.

K squared is equal to 1/2 multiplied by eight multiplied by K multiplied by sin 38.

If we simplify the expression on the right this gives k squared is equal to four k sign 38.

And then if we subtract four k sin 38 from both sides of the equation that will give us K squared subtract four k sin 38 is equal to zero.

We need to solve for k.

One way that we can do this is by factorising.

If we take out a factor of k, this will give k, brackets k subtract four sin 38 is equal to zero.

And this of course has two solutions.

This two expressions multiplied by one another are equal to zero.

So one of them must be zero.

So either k is equal to zero or k subtract four sin 38 is equal to zero.

And if k subtract four sin 38 is equal to zero, then k is equal to four sine 38.

And this gives 2.

46 centimetres to three significant figures.

So this is our answer.

Here's another question for you to try.

Pause the video to complete the task and resume the video once you've finished Here's the answer.

In this question we have to find the perimeter of the compound shape.

So, we have two missing lengths.

One of them is no problem.

If we split the shape up into a triangle and a rectangle as shown, then the side opposite the five centimetre side of the rectangle is also equal to five centimetres.

Then, we need to work out the area of the rectangle subtract this from the area of the compound shape.

That will then give you the area of the triangle.

Once you know this we can substitute into the area formula and solve as we have been doing And here's today's final question pause the video to complete the task and resume the video once you've finished.

Here's the answer.

This question was similar to the previous question, in that we need to find the perimeter.

So we need to finish by finding the total length around the outside of the shape.

It was really interesting question as well because when we partition this at shape into two triangles, we actually end up using both area formula that we know for triangles.

Both the area is equal to 1/2 multiplied by the base multiplied by the perpendicular height and the area is equal to 1/2 ab sin C.

That's all for this lesson.

Thanks for watching.