The length of the hypotenuse of a 30°-60°-90° triangle is 12. Find the perimeter.
I just took the test and the answer is in fact D. 18+6 sqrt 3
sides of standard 30-60-90 triangle are 1 , √3, 2
so the larger triangle has sides 6, 6√3 and 12
(multiplied by a factor of 6)
perimeter = 6 + 6√3 + 12 = 18 + 6√3
thanks
Well, well, well, if it isn't my old friend, the 30°-60°-90° triangle! So, we know that in this triangle, the hypotenuse is always twice as long as the shorter leg, and the longer leg is always the square root of 3 times the shorter leg.
Let's call the shorter leg "x". According to our trusty Pythagorean theorem, we can find the longer leg by multiplying x by the square root of 3, which in this case would be x√3.
Now, in this particular problem, we're given that the hypotenuse is 12. Since we know it's twice as long as the shorter leg, we can set up an equation:
2x = 12
x = 12/2
x = 6
So, the shorter leg, x, is 6. That means the longer leg would be 6√3.
Now, to find the perimeter, we just add up the lengths of all three sides. The shorter leg is 6, the longer leg is 6√3, and the hypotenuse is 12.
Perimeter = 6 + 6√3 + 12
And that, my friend, is the perimeter of our triangle. Now you can go forth and calculate with confidence!
To find the perimeter of a triangle, we need to know the lengths of all three sides.
In a 30°-60°-90° triangle, the side opposite the 30° angle is half the length of the hypotenuse, and the side opposite the 60° angle is √3 times the length of the side opposite the 30° angle.
So, if the length of the hypotenuse is 12, the side opposite the 60° angle is 12/2 = 6. And the side opposite the 30° angle is 6/√3 = 2√3.
Now, to find the perimeter, we sum up the lengths of all three sides:
Perimeter = hypotenuse + side opposite 60° angle + side opposite 30° angle
Perimeter = 12 + 6 + 2√3
Therefore, the perimeter of the triangle is 12 + 6 + 2√3 units.