Question Video: Finding the Length of an Arc given Its Circle’s Radius | Nagwa Question Video: Finding the Length of an Arc given Its Circle’s Radius | Nagwa

Question Video: Finding the Length of an Arc given Its Circle’s Radius Mathematics

Given that 𝑀𝐴 = 41, determine, to the nearest whole number, the length of arc 𝐴𝐶 and the area of the shaded region.

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Video Transcript

Given that 𝑀𝐴 equals 41, determine, to the nearest whole number, the length of arc 𝐴𝐶 and the area of the shaded region.

So what we’re gonna start by doing is finding the length of arc 𝐴𝐶. And the first thing we can notice to help us do this is the fact that the angle 𝑀𝐶𝑋 is equal to 90 degrees. That’s because we have a tangent and a radius. And these are perpendicular to each other. Well, therefore, the angle 𝐶𝑀𝐴 is also gonna be equal to 90 degrees because these are supplementary angles because we have a pair of parallel lines.

But why is this useful? Well, it’s useful because this tells us that the section we’re looking at is a quarter of our circle. And that’s because if we had 360 degrees, which is the whole of the circle, and divide it by four, we’d have 90 degrees. Okay, great. So now, let’s use this to find out the length of our arc 𝐴𝐶. And we can do that using the formula for the circumference of a circle. And this is that the circumference is equal to two 𝜋𝑟 or 𝜋𝑑. But what we know is that 𝑀𝐴 is equal to 41. So therefore, our radius is equal to 41.

So then what we’re gonna have is the fact that the length of arc 𝐴𝐶 is equal to two multiplied by 𝜋 multiplied by 41. And then this is divided by four cause as we’ve already ascertained, this is quarter of our circle. Well, this is gonna be equal to 20.5𝜋. And that’s because two multiplied by 41 is 82𝜋. Divide that by four, gives us 20.5𝜋. Well, if we calculate this, what we’re gonna get is 64.4026494. Well, if we check back at the question, we can see that it wants the answer left to the nearest whole number. So therefore, the answer for the length of arc 𝐴𝐶 is gonna be equal to 64.

Okay, great. So that’s the length of arc 𝐴𝐶. So now, what want to do is find the area of our shaded region. Now, to help us do this, what we’re gonna do is use the area of a circle is equal to 𝜋𝑟 squared and also that the area of a triangle is equal to a half the base times the height. So if we start off by finding the area of the triangle 𝐴𝐶𝑀. Well, this is gonna be equal to a half multiplied by 41 multiplied by 41. Or we could try a half multiplied by 41 squared. And that’s cause our base is 41 cause it’s our radius. And our height is also 41 because it’s also our radius. And this is equal to 840.5, which means we’ve now found the area of the triangle.

Okay, so what do we need to do next if we want to find the area of our shaded region? Well, next, what we want to do is find the area of our sector which I’ve outlined here in orange. And to work out the area of a sector, that’s gonna be equal to 𝜋 multiplied by 41 squared, that’s cause our 𝜋𝑟 squared, then divided by four because as we had ascertained earlier on, this is a quarter of a circle. And this is gonna be equal to 420.25𝜋.

Okay, great. So we’ve now got the area of our triangle and the area of the sector. So therefore, to find the area of the shaded region, what we want to do is take the area of the triangle away from the area of the sector. So therefore, the area of the shaded region is gonna be equal to 420.25𝜋 minus 840.5, which is gonna be equal to 479.7543127. Well, once again, we want this to the nearest whole number, which is gonna give an answer of 480. And then this will be unit squared.

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