What are the exact values of a and b?
Right triangle A B C is shown with labeled sides and one labeled angle. Side A B is the hypotenuse and it is labeled 10. Side B C is the height and it is labeled a. Side C A is the base and it is labeled b. The angle opposite side B C is 30 degrees.

A right triangle with one acute angle measuring 30° is the 30°-60°-90° "special" triangle. The ratios of its side lengths are well known to be ...

1 : √3 : 2

Side lengths

In the given triangle, the longest side is labeled 10, so is 5 times the measure of the longest side in the ratios given above. Multiplying those ratio values by 5, we have side lengths ...

a : b : c = 5 : 5√3 : 10

That is ...

a = 5

b = 5√3

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Additional comment

If you want to figure the side lengths using trigonometry, you have ...

a = c·sin(30°) = 10·(1/2) = 5

b = c·cos(30°) = 10·(√3/2) = 5√3

What are the exact values of a and b? Right triangle A B C is shown with labeled sides and one labeled angle. Side A B is the hypotenuse and it is labeled 10. Side B C is the height and it is labeled a. Side C A is the base and it is labeled b. The angle opposite side B C is 30 degrees.

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Side lengths

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What are the exact values of a and b?
Right triangle A B C is shown with labeled sides and one labeled angle. Side A B is the hypotenuse and it is labeled 10. Side B C is the height and it is labeled a. Side C A is the base and it is labeled b. The angle opposite side B C is 30 degrees.