### Video Transcript

Given that π΄π is 11 centimetres, find the length of the arc π΄π΅πΆ rounded to the nearest integer.

Firstly, letβs just be clear on the notation used in the question. I read this π΄π΅πΆ, with a circumflex or hat above it, as the arc π΄π΅πΆ which means the portion of the circumference of the circle that connects π΄ to π΅ to πΆ. Thatβs the length of this arc here which Iβve 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 π.

We recall that the circumference of a full circle can be found using the formula two ππ or ππ, where π represents the radius of the circle and π represents the diameter. To find an arc length, we take this full circumference of a circle, two ππ and multiply it by π 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 π΄π΅πΆ, we need to know two things, the radius of the circle and the angle π subtended by this arc at the centre.

Weβre given in the question that the length π΄π is 11 centimetres. And π΄π is a radius of the circle because it connects the centre, π, with a point on the circumference, π΄. So we found the radius of the circle. Letβs consider angle π. Looking at triangle π΄πΆπ, we can see that weβve been given one angle in this triangle, the angle of 42 degrees. Now, the line ππΆ is also a radius of the circle. It connects the centre, π, with the point πΆ which is on the circumference. And so itβs also equal to 11 centimetres which means that the triangle π΄πΆπ is an isosceles triangle with the line π΄πΆ as its base.

Isosceles triangles have two equal angles. So angle π΄πΆπ is equal to angle πΆπ΄π. Theyβre 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 π, 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 weβve found the angle subtended by this arc at the centre of the circle.

We can now substitute the values of π and π into our formula for arc length. And we have that the length of the arc π΄π΅πΆ 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. Itβs a four. And as this is less than five, this tells us that weβre 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. Weβve found that the length of the arc π΄π΅πΆ rounded to the nearest integer is 18 centimetres.