Video: Pack 4 • Paper 3 • Question 14

Pack 4 • Paper 3 • Question 14

05:38

Video Transcript

𝐴𝐵, 𝐵𝐶, and 𝐶𝐷 are sides of a regular 10-sided polygon. And 𝐶𝐷 and 𝐶𝐸 are sides of a regular octagon. Work out the size of angle 𝐶𝐸𝐵. You must show all of your working out.

First of all, we’re actually gonna mark on to the diagram some of the things that we know. First of all, because we actually have 𝐶𝐷 as a shared side — so it’s a side of the regular 10-sided polygon and it’s also a side of the regular octagon — so therefore we can actually say that the sides 𝐵𝐶 and 𝐶𝐸 must also be the same, because we’ve already said, like I said, that 𝐶𝐷 is a shared side in both of the polygons.

Okay, great! because of this, we actually know that the triangle 𝐵𝐶𝐸 is actually going to be an isosceles triangle. And this is gonna be really helpful a bit later on in the question. To enable us to actually find the angle 𝐶𝐸𝐵, which is what the question is looking for, we need to first of all find out the angle 𝐵𝐶𝐸. And the way we can actually do that is by splitting it in two, because actually what we’ve got here are two exterior angles. So what we actually have is the exterior angle of our 10-sided polygon, which is called a decagon, and also the exterior angle of the regular octagon, so our eight-sided shape.

Okay, so how do we find out what the exterior angle is going to be? Well, there are actually a couple of ways we can solve this using different relationships and formula. So first of all, we know that the exterior angle of a regular polygon is equal to 360 degrees divided by 𝑛, where 𝑛 is the number of sides of the polygon.

The key word that allows us to use this is actually regular because we do know that the exterior angle of any polygon add up to 360 degrees. However, we won’t be able to calculate the individual values of those exterior angles if it wasn’t a regular polygon. But because it’s a regular polygon, we can actually do that, because we can just divide 360 degrees by 𝑛, the number of sides.

Okay, so that’s one method. The next method is actually to calculate one of the interior angles. And you can calculate an interior angle by 180 multiplied by 𝑛 minus two over 𝑛. And that’s where 𝑛 again is the size of our polygon. And we can use that to solve this kind of problem, because actually we know another relationship and that’s the exterior plus interior angles add up to 180 degrees. And this is because their angles are on a straight line.

Okay, great! So we’ve got some different ways to solve this. First of all, I’m gonna start using the first formula to solve it. So I’m gonna start with the exterior angle of our decagon. And I’ve actually used 𝐸𝐷 to represent that. So we can say that 𝐸𝐷 is equal to 360 divided by 10, because it’s 10 sides, which gives us an angle of 36 degrees.

And now we’re gonna move on to the exterior angle of the octagon. And to calculate the exterior angle of our octagon, it’s gonna be 360 degrees divided by eight, which gives an angle of 45 degrees.

Now as I said before, we can actually check this by using the other method. So if we do, we’ve got the interior angle of 𝐷, so the interior angle of our decagon, is equal to 180 multiplied by 10 minus two over 10, which is equal to 1440 divided by 10, which gives us 144 degrees. So that’s our interior angle.

So therefore, using our final relationship that an exterior plus an interior angle equals 180 degrees, we can state that the exterior angle of the decagon is gonna be equal to 180 minus 144, which gives us 36 degrees.

So first of all, we can say that the interior angle of our octagon is equal to 180 multiplied by eight minus two divided by eight, which gives us 1080 divided by eight, which is equal to 135 degrees. Okay, great! So again now we can move on and find the exterior angle of the octagon. And again, we use our final relationship that exterior plus interior equals 180 degrees. So we can say that the exterior angle of our octagon is equal to 180 minus 135, which gives us an angle of 45 degrees, which again, great, fits the angle that we got using the first method.

Okay, so now let’s move on and actually work out the size of angle 𝐶𝐸𝐵. So what I’ve actually done is drawn a sketch of the triangle that we’re using here. So we’re trying to find the angle 𝐶𝐸𝐵. But we’ve just found out the angle 𝐵𝐶𝐸, because the angle 𝐵𝐶𝐸 is equal to 36 degrees plus 45 degrees, cause they are exterior angles that we calculated. So this is gonna give us an angle of 81 degrees.

Okay, great! So now what’s the next step? Well, the next step is to actually use the relationship that we found very early on. And that’s that this is actually an isosceles triangle. So therefore, 𝐶𝐸𝐵 and 𝐶𝐵𝐸 are actually the same. So they’re equal to each other. So therefore, we can say that 𝐶𝐸𝐵 plus 𝐶𝐵𝐸 is gonna be equal to 180 minus 81, cause that’s just the angles in a triangle adding up to 180 degrees. And this gives an answer of 99 degrees.

So great! We know that if we add those two angles together, it’s gonna give us 99 degrees. So therefore, angle 𝐶𝐸𝐵 is gonna be equal to 99 divided by two. That’s because as it’s an isosceles triangle, we’ve already said 𝐶𝐸𝐵 and 𝐶𝐵𝐸 are gonna be the same. So then we get our final answer. And that’s that 𝐶𝐸𝐵 is equal to 49.5 degrees, cause 99 divided by two gives us 49.5 degrees.

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