### Video Transcript

Given that ๐๐ด equals 41, determine, to the nearest whole number, the length of arc ๐ด๐ถ and the area of the shaded region.

So what weโre gonna start by doing is finding the length of arc ๐ด๐ถ. And the first thing we can notice to help us do this is the fact that the angle ๐๐ถ๐ is equal to 90 degrees. Thatโs because we have a tangent and a radius. And these are perpendicular to each other. Well, therefore, the angle ๐ถ๐๐ด 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, itโs useful because this tells us that the section weโre looking at is a quarter of our circle. And thatโs because if we had 360 degrees, which is the whole of the circle, and divide it by four, weโd have 90 degrees. Okay, great. So now, letโs use this to find out the length of our arc ๐ด๐ถ. And we can do that using the formula for the circumference of a circle. And this is that the circumference is equal to two ๐๐ or ๐๐. But what we know is that ๐๐ด is equal to 41. So therefore, our radius is equal to 41.

So then what weโre gonna have is the fact that the length of arc ๐ด๐ถ is equal to two multiplied by ๐ multiplied by 41. And then this is divided by four cause as weโve already ascertained, this is quarter of our circle. Well, this is gonna be equal to 20.5๐. And thatโs because two multiplied by 41 is 82๐. Divide that by four, gives us 20.5๐. Well, if we calculate this, what weโre 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 ๐ด๐ถ is gonna be equal to 64.

Okay, great. So thatโs the length of arc ๐ด๐ถ. So now, what want to do is find the area of our shaded region. Now, to help us do this, what weโre gonna do is use the area of a circle is equal to ๐๐ 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 ๐ด๐ถ๐. 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 thatโs cause our base is 41 cause itโs our radius. And our height is also 41 because itโs also our radius. And this is equal to 840.5, which means weโve 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 Iโve outlined here in orange. And to work out the area of a sector, thatโs gonna be equal to ๐ multiplied by 41 squared, thatโs cause our ๐๐ 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 420.25๐.

Okay, great. So weโve 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 420.25๐ 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.