# Question Video: Finding the Unknown Lengths in a Triangle given the Other Sidesβ Lengths in a Similar Triangle Mathematics

Triangles π΄π΅πΆ and π·πΈπΉ are similar. What is the length of πΈπΉ?

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### Video Transcript

Triangles π΄π΅πΆ and π·πΈπΉ are similar. What is the length of πΈπΉ?

We know that in any similar triangles, the corresponding angles are equal. In this case, angle π΄ is equal to angle π·, angle π΅ equals angle πΈ, and angle πΆ is equal to angle πΉ. The three angles in any triangle sum to 180 degrees. This means that from triangle π΄π΅πΆ, three π₯ plus two π₯ plus π₯ is equal to 180. Simplifying the left-hand side gives us six π₯. We can then divide both sides of this by six, giving us a value of π₯ equal to 30.

The three angles in triangle π΄π΅πΆ are 90 degrees, 60 degrees, and 30 degrees. This means that the corresponding angles in triangle π·πΈπΉ will also be 90 degrees, 60 degrees, and 30 degrees. This means that triangle π·πΈπΉ is a right-angled triangle. And we can use right angle trigonometry and our knowledge of special trig angles to calculate the length πΈπΉ.

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 π·πΈ is opposite this. The length π·πΉ 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 π is equal to the opposite over the hypotenuse. Substituting in our values gives us sin of 30 is equal to 7.8 over π₯. 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 π₯.

Cross Multiplying at this stage means we are multiplying both sides by π₯ and by two. This gives us π₯ is equal to 7.8 multiplied by two. This, in turn, is equal to 15.6. The length πΈπΉ 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.