A right triangle has angle 30° and adjacent side 8. Find the hypotenuse
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We are given a right triangle with a 30 degree angle and an adjacent side of length 8. Our goal is to find the length of the hypotenuse. Let's visualize this triangle with the known information clearly labeled.
In any right triangle, the sum of angles is 180 degrees. With one angle being 90 degrees and another 30 degrees, the third angle must be 60 degrees. This makes it a special 30-60-90 triangle. These triangles have fixed side ratios: if the shortest side (opposite to 30 degrees) is of length 1, then the side opposite to 60 degrees is square root of 3, and the hypotenuse is 2.
To solve this problem using trigonometry, we'll use the cosine ratio. Cosine of an angle in a right triangle is defined as the length of the adjacent side divided by the length of the hypotenuse. For our 30 degree angle, this gives us cosine of 30 degrees equals 8 over h. We know from trigonometric values that cosine of 30 degrees equals square root of 3 over 2.
Now we'll solve the equation step by step. We have square root of 3 over 2 equals 8 over h. Cross multiplying gives us h times square root of 3 equals 16. Dividing both sides by square root of 3 gives h equals 16 over square root of 3. To rationalize the denominator, we multiply both numerator and denominator by square root of 3, resulting in h equals 16 square root of 3 over 3.
Let's solve this using the properties of 30-60-90 triangles. We know the side ratios are 1 to square root of 3 to 2. In our triangle, the side adjacent to the 30 degree angle (opposite to 60 degrees) corresponds to the square root of 3 part of the ratio. If we let the side opposite to 30 degrees be x, then the side opposite to 60 degrees is x times square root of 3, which equals 8. Solving for x gives us 8 over square root of 3. The hypotenuse is 2x, which equals 16 over square root of 3, or 16 square root of 3 over 3 after rationalizing.
Let's verify our answer. If the hypotenuse is 16 square root of 3 over 3, then cosine of 30 degrees should equal 8 divided by this value. This simplifies to square root of 3 over 2, which is the correct value for cosine of 30 degrees. As a decimal approximation, this is about 9.24. Both our trigonometric and special triangle methods yielded the same result, confirming that the hypotenuse is 16 square root of 3 over 3.