Given that 𝐴𝑀 is 11 centimetres, find the length of the arc 𝐴𝐵𝐶 rounded to the nearest integer.
Firstly, let’s just be clear on the notation used in the question. I read this 𝐴𝐵𝐶, with a circumflex or hat above it, as the arc 𝐴𝐵𝐶 which means the portion of the circumference of the circle that connects 𝐴 to 𝐵 to 𝐶. That’s the length of this arc here which I’ve marked in orange. Now, an arc is always subtended by an angle at the centre of the circle. This is the angle formed by the radii which connect the two endpoints of the arc to the centre of the circle and is often referred to using the Greek letter 𝜃.
We recall that the circumference of a full circle can be found using the formula two 𝜋𝑟 or 𝜋𝑑, where 𝑟 represents the radius of the circle and 𝑑 represents the diameter. To find an arc length, we take this full circumference of a circle, two 𝜋𝑟 and multiply it by 𝜃 over 360, which is the fraction of the full circumference of the circle that this arc length represents. So, to work out the length of arc 𝐴𝐵𝐶, we need to know two things, the radius of the circle and the angle 𝜃 subtended by this arc at the centre.
We’re given in the question that the length 𝐴𝑀 is 11 centimetres. And 𝐴𝑀 is a radius of the circle because it connects the centre, 𝑀, with a point on the circumference, 𝐴. So we found the radius of the circle. Let’s consider angle 𝜃. Looking at triangle 𝐴𝐶𝑀, we can see that we’ve been given one angle in this triangle, the angle of 42 degrees. Now, the line 𝑀𝐶 is also a radius of the circle. It connects the centre, 𝑀, with the point 𝐶 which is on the circumference. And so it’s also equal to 11 centimetres which means that the triangle 𝐴𝐶𝑀 is an isosceles triangle with the line 𝐴𝐶 as its base.
Isosceles triangles have two equal angles. So angle 𝐴𝐶𝑀 is equal to angle 𝐶𝐴𝑀. They’re both equal to 42 degrees. We could express this reasoning by writing down that the triangle is isosceles because it has been formed by two radii. We also know that the angles in any triangle sum to 180 degrees. So we can work out the third angle in this triangle, angle 𝜃, by subtracting the two angles of 42 degrees from 180 degrees. Two lots of 42 are 84. And subtracting this form 180 gives 96. So we’ve found the angle subtended by this arc at the centre of the circle.
We can now substitute the values of 𝜃 and 𝑟 into our formula for arc length. And we have that the length of the arc 𝐴𝐵𝐶 is equal to 96 over 360 multiplied by two multiplied by 𝜋 multiplied by 11. We can use a calculator to evaluate this. And it gives 18.43067 continuing.
The question asks us to give our answer to the nearest integer. So we look at our deciding number which is the first digit after the decimal point. It’s a four. And as this is less than five, this tells us that we’re going to be rounding down. So we round down to 18 and include the units for this arc length which are centimetres because the radius was given in centimetres. We’ve found that the length of the arc 𝐴𝐵𝐶 rounded to the nearest integer is 18 centimetres.