Formula for Angles to Diagonals

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The formula for angles to diagonals is something that you can do while working with shapes like the rhombus. Find out the formula for angles to diagonals with help from an experienced math professional in this free video clip.

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Hi, I'm Drew Moyer. And this is a formula for angles to diagonals. Let's suppose I'm given this rhombus and I know the length of the sides, as well as the measure of the angles. But now I'm told that I need to find the length of the diagonals. Well, I could start with three things that I know about the diagonals of a rhombus. One is that they bisect the angles that they originate from. So, if I know that this angle is 60, I know that this angle must be 60. Which means when bisected, I know that each one of these angles is 30. And now, same thing for up here, I know that this angle in entirety is 120. So, I know that when bisected, it must be 60. The next thing I know about angles, about diagonals of a rhombus, is that they bisect each other. Meaning that they cut each other in half. So, I know that this side right here, must equal this side over here. And this side over here must equal this side over here. And the final thing I know is that they bisect each other at a 90 degree angle. Which means that all these angles inside are right angles. And now, as you can see, I have four congruent right triangles. And not only are they right triangles, but they're all 30, 60, 90 triangles. Which means that I can tell a lot of things about the sides, without having to use Pythagorean Theorem. For example, let's look at this top triangle. I know that the hypotenuse is four. And I also know that the side across from the 30 degree angle is equal to half the length of the hypotenuse. Which means that it's two. And since this side is congruent to this side, this must also be two. So, I have the length of our short diagonal, it's two plus two or four. Now, for the longer one, I'm going to turn to the 60 degree angle. I know that the side across from the 60 degree angle or the medium side in a 30, 60, 90 triangle, must be the short side times the square root of three. So, I know that this side must be two root three, and the same thing down here for this side because they're congruent, two root three. So, the length of the longer diagonal then is two root three, plus two root three. Which is equal to four root thee. So now, I have the length of both diagonals. So, I'm Drew Moyer, and this is a formula for angles to diagonals.

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