A right triangle has angle 30° and adjacent side 10. Find the hypotenuse
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We are given a right triangle with a 30 degree angle and an adjacent side of length 10. Our goal is to find the length of the hypotenuse. Let's visualize this triangle and label all the known information.
Let's review the names of the sides in a right triangle relative to a given angle. The hypotenuse is always the longest side, opposite the right angle. The adjacent side is the one that forms the angle with the hypotenuse. The opposite side is across from the angle. In our triangle, with the 30 degree angle at A, the side of length 10 is adjacent to the angle.
Trigonometric ratios are fundamental relationships in right triangles. The sine of an angle is the ratio of the opposite side to the hypotenuse. The cosine is the ratio of the adjacent side to the hypotenuse. The tangent is the ratio of the opposite side to the adjacent side. Since we know the adjacent side and want to find the hypotenuse, we'll use the cosine ratio.
Now we'll apply the cosine ratio to solve for the hypotenuse. We write cos(30 degrees) equals adjacent over hypotenuse, which gives us cos(30 degrees) equals 10 over h. We know that cos(30 degrees) equals square root of 3 over 2. Substituting this value, we get the equation h equals 10 divided by square root of 3 over 2.
Let's solve for h step by step. We have h equals 10 divided by square root of 3 over 2. This is the same as 10 times 2 over square root of 3, which equals 20 over square root of 3. To rationalize the denominator, we multiply both numerator and denominator by square root of 3, giving us 20 square root of 3 over 3. The decimal approximation of this value is approximately 11.55.
Our triangle is actually a special 30-60-90 right triangle. These triangles have a consistent side ratio pattern. The sides are always in the ratio 1 to square root of 3 to 2, where 1 is the short leg opposite the 30 degree angle, square root of 3 is the long leg opposite the 60 degree angle, and 2 is the hypotenuse. Since our short leg is 10, which is 10 times 1, our hypotenuse must be 10 times 2, which equals 20. This gives us a quick way to solve this type of problem.