Can someone help me with 30-60-90 triangles? I know I should remember this from geometry class, but I'm a little confused. What's the most efficient way to solve them?
When working with 30-60-90 triangles, it's important to recognize the ratios between the side lengths. The ratios, based on the angles of the triangle, are as follows:
1. The side opposite the 30-degree angle is the shortest side, and we'll call it x.
2. The side opposite the 60-degree angle is √3 times longer than the side opposite the 30-degree angle, so its length will be x√3.
3. The side opposite the 90-degree angle (the hypotenuse) is twice the length of the side opposite the 30-degree angle, so its length will be 2x.
Given one side length of a 30-60-90 triangle, you can easily find the other two sides using these ratios.
Here's an example to illustrate the process:
Suppose you have a 30-60-90 triangle with a hypotenuse length of 10. Using the ratios above, you can find the other two side lengths:
1. Since the hypotenuse is 2x, and it has a length of 10, then x = 10/2 = 5. This gives us the side length opposite the 30-degree angle, which is 5.
2. To find the side length opposite the 60-degree angle, multiply x by √3: 5√3.
Now you have all the side lengths: the side opposite 30 degrees is 5, the side opposite 60 degrees is 5√3, and the side opposite 90 degrees (the hypotenuse) is 10.
Remembering these ratios and applying them accordingly is the most efficient way to solve 30-60-90 triangles.
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