Question Video: Finding the Unknown Lengths in a Triangle given the Other Sides’ Lengths in a Similar Triangle Mathematics

Video Transcript

Triangles 𝐴𝐵𝐶 and 𝐷𝐸𝐹 are similar. What is the length of 𝐸𝐹?

We know that in any similar triangles, the corresponding angles are equal. In this case, angle 𝐴 is equal to angle 𝐷, angle 𝐵 equals angle 𝐸, and angle 𝐶 is equal to angle 𝐹. The three angles in any triangle sum to 180 degrees. This means that from triangle 𝐴𝐵𝐶, three 𝑥 plus two 𝑥 plus 𝑥 is equal to 180. Simplifying the left-hand side gives us six 𝑥. We can then divide both sides of this by six, giving us a value of 𝑥 equal to 30.

The three angles in triangle 𝐴𝐵𝐶 are 90 degrees, 60 degrees, and 30 degrees. This means that the corresponding angles in triangle 𝐷𝐸𝐹 will also be 90 degrees, 60 degrees, and 30 degrees. This means that triangle 𝐷𝐸𝐹 is a right-angled triangle. And we can use right angle trigonometry and our knowledge of special trig angles to calculate the length 𝐸𝐹.

We recall the acronym SOHCAHTOA, which helps us find the sine, cosine, and tangent ratios in right-angled triangles. The longest side of a right-angled triangle is opposite the right angle. This is known as the hypotenuse. If we focus on the 30-degree angle, we see that length 𝐷𝐸 is opposite this. The length 𝐷𝐹 is adjacent or next to the 30-degree angle and the right angle.

As we are dealing with the opposite and hypotenuse, we will use the sine ratio. This states that sin of 𝜃 is equal to the opposite over the hypotenuse. Substituting in our values gives us sin of 30 is equal to 7.8 over 𝑥. sin of 30 degrees is one of our special trig angles. It is equal to one-half. This means that one-half is equal to 7.8 over 𝑥.

Cross Multiplying at this stage means we are multiplying both sides by 𝑥 and by two. This gives us 𝑥 is equal to 7.8 multiplied by two. This, in turn, is equal to 15.6. The length 𝐸𝐹 is equal to 15.6 centimeters. It is worth noting here that this triangle follows a general pattern. If we have a right-angled triangle where another angle is equal to 30 degrees, then the opposite is always half the length of the hypotenuse.