Find the perimeter of triangle 𝐴𝐵𝐶.
We’re asked to find the perimeter here. That’s the distance all the way round the outside of a shape. The triangle here is this smaller triangle, 𝐴𝐵𝐶. In order to find the perimeter of triangle 𝐴𝐵𝐶, 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 𝐷𝐶, which is given as 14.1 centimeters, will be the same as the line 𝐶𝐵 and the line 𝐴𝐶. So, these are also both 14.1 centimeters. There’s no immediate way to find the length of this line 𝐴𝐵, so let’s have a look at the angles that we’re given.
Let’s consider this triangle 𝐴𝐶𝐷. Now, we’re told that two lengths are the same, which means that this triangle 𝐴𝐶𝐷 is an isosceles triangle. It also means that we’ll have two equal angles. We can then say that this angle 𝐷𝐴𝐶 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 𝐴𝐶𝐷. We would calculate 180 degrees subtract 30 degrees subtract 30 degrees. And as that’s the same as subtracting 60 degrees, we’d be left with 120 degrees.
Now, we’re still really interested in triangle 𝐴𝐵𝐶. So, let’s see what we can find out about this triangle. If we use the fact that we have this straight line 𝐷𝐵 and the fact that angles on a straight line add up to 180 degrees, we can find this angle 𝐴𝐶𝐵. This angle will be equal to 180 degrees subtract 120 degrees, which is 60 degrees. We may feel at this point that we’re still no closer to finding the length of 𝐴𝐵, but let’s consider the type of triangle that 𝐴𝐵𝐶 is.
Just like our previous triangle 𝐴𝐶𝐷, we could say that 𝐴𝐵𝐶 is also an isosceles triangle, so it will also have two equal angles. Angle 𝐶𝐴𝐵 will be the same size as angle 𝐴𝐵𝐶. 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, 𝐴𝐵𝐶? It’s 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 𝐴𝐵. It’s also 14.1 centimeters.
We can now find the perimeter of triangle 𝐴𝐵𝐶 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.