### Video Transcript

Find the perimeter of triangle ๐ด๐ต๐ถ.

Weโre asked to find the perimeter here. Thatโs the distance all the way round the outside of a shape. The triangle here is this smaller triangle, ๐ด๐ต๐ถ. In order to find the perimeter of triangle ๐ด๐ต๐ถ, we need to know the length of all three of the sides of this triangle. If we look at the line markings, we can see that the length ๐ท๐ถ, which is given as 14.1 centimeters, will be the same as the line ๐ถ๐ต and the line ๐ด๐ถ. So, these are also both 14.1 centimeters. Thereโs no immediate way to find the length of this line ๐ด๐ต, so letโs have a look at the angles that weโre given.

Letโs consider this triangle ๐ด๐ถ๐ท. Now, weโre told that two lengths are the same, which means that this triangle ๐ด๐ถ๐ท is an isosceles triangle. It also means that weโll have two equal angles. We can then say that this angle ๐ท๐ด๐ถ must also be 30 degrees. We should remember that the angles in a triangle add up to 180 degrees, which means that we can find the size of this angle ๐ด๐ถ๐ท. We would calculate 180 degrees subtract 30 degrees subtract 30 degrees. And as thatโs the same as subtracting 60 degrees, weโd be left with 120 degrees.

Now, weโre still really interested in triangle ๐ด๐ต๐ถ. So, letโs see what we can find out about this triangle. If we use the fact that we have this straight line ๐ท๐ต and the fact that angles on a straight line add up to 180 degrees, we can find this angle ๐ด๐ถ๐ต. This angle will be equal to 180 degrees subtract 120 degrees, which is 60 degrees. We may feel at this point that weโre still no closer to finding the length of ๐ด๐ต, but letโs consider the type of triangle that ๐ด๐ต๐ถ is.

Just like our previous triangle ๐ด๐ถ๐ท, we could say that ๐ด๐ต๐ถ is also an isosceles triangle, so it will also have two equal angles. Angle ๐ถ๐ด๐ต will be the same size as angle ๐ด๐ต๐ถ. Using the fact that the angles in a triangle add up to 180 degrees, these two angles must therefore add together to give 120 degrees, meaning that both of these angles will be 60 degrees. So, now, what can we say about this triangle, ๐ด๐ต๐ถ? Itโs not just isosceles; it is, in fact, an equilateral triangle.

We know this because in an equilateral triangle, all the internal angles are 60 degrees. And importantly for us, we know that all the sides in an equilateral triangle are the same length. This means that we now know the length of ๐ด๐ต. Itโs also 14.1 centimeters.

We can now find the perimeter of triangle ๐ด๐ต๐ถ by adding 14.1, 14.1, and 14.1 or alternatively three multiplied by 14.1. This will give us our final answer of 42.3 centimeters.