A right triangle has angle 30° and adjacent side 9. Find the hypotenuse
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We are given a right triangle with a 30 degree angle and an adjacent side of length 9. We need to find the hypotenuse.
Since the sum of angles in a triangle is 180 degrees, and we have a right angle and a 30 degree angle, the third angle must be 60 degrees. This makes it a special 30-60-90 triangle.
To find the hypotenuse, we can use the cosine ratio. Cosine of an angle equals adjacent side over hypotenuse. So cos(30 degrees) equals 9 over hypotenuse. We know that cos(30 degrees) equals square root of 3 over 2.
Now we set up the equation using our known values. We have cosine of 30 degrees equals 9 divided by the hypotenuse. Since cosine of 30 degrees is square root of 3 over 2, we substitute that in.
To solve for the hypotenuse, we cross multiply to get hypotenuse times square root of 3 equals 18. Dividing both sides by square root of 3 gives us hypotenuse equals 18 over square root of 3. Rationalizing the denominator, we multiply numerator and denominator by square root of 3, resulting in hypotenuse equals 6 times square root of 3.
Let's verify our answer. We substitute back into the cosine ratio: cosine of 30 degrees equals 9 divided by 6 times square root of 3. Simplifying, we get 9 times square root of 3 over 18, which equals square root of 3 over 2. This confirms our answer is correct. The hypotenuse is 6 times square root of 3, which is approximately 10.39 units.