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### Course: Class 10 (Old)>Unit 8

Lesson 3: Trigonometric ratios of some specific angles

# Special right triangles proof (part 1)

Learn how to prove the ratios between the sides of a 30-60-90 triangle. Created by Sal Khan.

## Want to join the conversation?

• this is a little confusing, especially at the beginning. help
• I would like to help, but please be more specific on what is confusing to you.
• At , Sal mentions taking the "principal root" of something. What is a principal root, and how is it different than square root? Thanks!
• The square root of a number is both positive and negative:
√9 = ±3
Because both 3 and -3 squared give you 9. The principal root is just the positive answer. So the principal square root of 9 is 3.
• how did u get x squared over four?
• (x/2) squared means you square both the numerator and the denominator, so that's where the x^2 over 4 comes from.
• When writing out the Pythagorean theorem, I'm a bit confused about why I'm not allowed to take the square root of each term.
So, a^2 + b^2 = c^2
Take the root of both sides
a + b = c
• It is quite tempting to think that the square root of (a^2 + b^2) is a+b, but this simply does not work. The easiest way to see this is to plug in numbers for a and b, such as a=3 and b=4.
sqrt(3^2+4^2) = sqrt(9+16) = sqrt(25) = 5.
However, 3+4 = 7, which is not 5.
• I know that an `=` with a `~` above it means congruent, but what does just a `~` mean in math? Or does it even have to do with math? And if it is used in math, what would you use it for, and when does it come in, in math (like which grade?).
• Often, ~ is used to mean "approximately equal to".
e.g. 1.7498 ~ 1.75

It would typically be introduced around the same time you learn about decimals and fractions (your school may vary).
• : where does that 4 come from?
• That's because (x/2)^2 = (x^2)/(2^2) = (x^2)/4. When you square a fraction, you square both the numerator and the denominator. Pay attention to parentheses, and you'll be fine.

If you're asking about that 4 he glued to x^2 on the right side, then don't forget that he immediately divided 4x^2 by 4, which is the same as x^2 since 4/4 = 1. This is a useful trick to help you add/subtract fractions without multiplying everything (in both sides) by what's in the denominator in order to get rid of fractions. He'd have to divide by 4 again, anyway, in order to solve for BD.
• could you have a right triangle which has the following: a right angle, an angle of 89 and an angle of 1
• Yep! Just make sure that the angles all add up to 180º. Hope this helps!
—CT-2/002-24
• Shouldn't the side opposite the 60 degrees angle be double the side opposite the 30 degrees angle? Logically, wouldn't it make sense that an angle that is double the size of another angle, should have a corresponding segment that is double the segment made by the smaller angle? Why is this not the case?
• You are suggesting that the sides should be in proportion to the angles but in fact there is no reason this should be true.
Here is one argument against it.
If the sides were in proportion to the angles, then the hypotenuse (the side opposite the 90 degree angle) would be triple the side opposite the 30 degree angle. The sides would be 1, 2, 3 or 2, 4, 6, etc. This is clearly impossible since the third side has to be shorter than the sum of the other 2 sides, since the shortest side is a straight line. Another reason we know this isn't true is because it wouldn't satisfy the pythagorean theorem 1^2 + 2^2 doesn't = 3^2.
Hope this helps.