The length of the hypotenuse of a 30 60 90 triangle is 16. What is the perimeter?
I'm just learning about this. Can someone explain?
the sides are in the ratio 1:√3:2 so if the hypotenuse is 16, then the sides are
8,8√3,16 and the perimeter is 24+8√3
Thank you Steve!
Well, well, well, it seems like you're in a right-angled pickle! Don't worry, I'm here to help.
For a 30-60-90 triangle, the lengths of the sides are always in a special ratio. The sides opposite the angles are usually called the opposite or opposite sides, so in this case, the side opposite the 30-degree angle is the shortest side (let's call it x), the side opposite the 60-degree angle is √3 times that length (√3x), and the hypotenuse (opposite the right angle) is twice the length of the shortest side (2x).
You mentioned that the hypotenuse is 16, so we can set up an equation based on the 30-60-90 relationship:
2x = 16
To find the value of x, we divide both sides by 2:
x = 8
Now that we know the length of the shortest side, we can find the other sides:
The side opposite the 60-degree angle (√3x) would be (√3 * 8) = 8√3,
And the side opposite the 30-degree angle (x) would be 8.
To find the perimeter, we add up the lengths of all three sides:
Perimeter = x + 8√3 + 8
Perimeter = 8 + 8√3 + 8
Perimeter = 16 + 8√3
So, the perimeter of the triangle is 16 + 8√3. There you have it, a right-angled triangle with a punchy perimeter!
Sure! A 30-60-90 triangle is a special type of right triangle where the angles measure 30 degrees, 60 degrees, and 90 degrees. In this triangle, the side lengths follow a specific ratio.
The ratio of the sides in a 30-60-90 triangle is:
- The side opposite the 30-degree angle is always half the length of the hypotenuse.
- The side opposite the 60-degree angle is always √3 times the length of the shorter side.
- The length of the hypotenuse is always twice the length of the shorter side.
To find the perimeter of the triangle, we need to add up the lengths of all three sides.
Given that the length of the hypotenuse is 16, we can find the lengths of the other two sides using the ratio mentioned earlier:
- The length of the shorter side (opposite the 30-degree angle) is 16/2 = 8.
- The length of the longer side (opposite the 60-degree angle) is 8 * √3.
Now, to find the perimeter, we add up all three sides:
Perimeter = 8 + 8 * √3 + 16
Simplifying the expression, the perimeter is:
Perimeter = 8 + 8 * √3 + 16
Perimeter = 24 + 8 * √3
Sure! I'd be happy to explain how to solve this problem.
First, let's understand what a 30 60 90 triangle is. A 30 60 90 triangle is a right triangle with angles measuring 30 degrees, 60 degrees, and 90 degrees. The ratio of the sides in a 30 60 90 triangle is always 1:√3:2, where the side opposite the 30-degree angle is half the length of the hypotenuse, and the side opposite the 60-degree angle is √3 times the length of the shorter leg.
In this case, we are given that the length of the hypotenuse is 16. Since the side opposite the 30-degree angle is half the length of the hypotenuse, we can calculate its length by dividing the length of the hypotenuse by 2. Therefore, the shorter leg is 16/2 = 8.
Now, to find the length of the longer leg, we need to multiply the length of the shorter leg (8) by √3. So the longer leg is 8√3.
To find the perimeter of the triangle, we need to add up the lengths of all three sides. The perimeter is given by the formula: perimeter = (shorter leg) + (longer leg) + (hypotenuse).
Using the values we calculated, the perimeter is:
8 + 8√3 + 16 = 8 + 8√3 + 16 = 24 + 8√3.
Therefore, the perimeter of the triangle is 24 + 8√3.