The length of the hypotenuse of a 30-60-90 triangle is 12. Find the perimeter.
the ratio of sides of the 30-60-90 triangle
= 1 : √3 : 2
= 6 :6√3: 12
so perimeter = 6 + 6√3 + 12 = 18 + 6√3
To find the perimeter of a 30-60-90 triangle, we need to know the lengths of the other two sides.
In a 30-60-90 triangle, the ratio of the sides is:
Shortest side : Medium side : Longest side = 1 : √3 : 2.
Given that the length of the hypotenuse is 12, we can set up the following equation to find the length of the other two sides:
√3x = 12,
where x represents the length of the medium side.
To solve for x, we divide both sides of the equation by √3:
x = 12/√3.
To rationalize the denominator, we multiply the numerator and denominator by √3:
x = (12/√3) * (√3/√3)
= 12√3/3
= 4√3.
Therefore, the length of the medium side is 4√3.
We can find the length of the shortest side by using the ratio:
Shortest side = (1/2) * length of hypotenuse
= (1/2) * 12
= 6.
Now we can calculate the perimeter by adding the lengths of all three sides:
Perimeter = Shortest side + Medium side + Hypotenuse
= 6 + 4√3 + 12
= 18 + 4√3.
Therefore, the perimeter of the 30-60-90 triangle is 18 + 4√3.
To find the perimeter of a triangle, we need to find the lengths of all three sides.
A 30-60-90 triangle is a special right triangle where the angles are in the ratio 1:2:2. In this type of triangle, the length of the shorter leg (opposite the 30-degree angle) is half the length of the hypotenuse, and the length of the longer leg (opposite the 60-degree angle) is √3 times the length of the shorter leg.
Given that the length of the hypotenuse is 12, we can find the lengths of the other two sides.
The shorter leg is half the length of the hypotenuse, so it is 12/2 = 6.
The longer leg is √3 times the length of the shorter leg, so it is √3 * 6 = 6√3.
Now, to find the perimeter, we need to add up the lengths of all three sides:
Perimeter = hypotenuse + shorter leg + longer leg
Perimeter = 12 + 6 + 6√3
Perimeter = 18 + 6√3
Therefore, the perimeter of the triangle is 18 + 6√3 units.