The length of the hypotenuse of a 30 degrees, -60 degrees, and -90 degrees triangle is 4. What is the perimeter?
4+12 sqrt 3
6+2 sqrt 3
2+6 sqrt 3
12+ 4 sqrt 3
As part of your repertoire of essential math facts, your should know the sides of a
30-60-90 triangle are 1 : √3 : 2, where 2 is the hypotenuse.
so the perimeter would be 3 + √3
Your hypotenuse is 4, twice the above, so the perimeter would be 6 + 2√3
To find the perimeter of a triangle, we need to find the lengths of all three sides.
In a 30-60-90 triangle, the side opposite the 30-degree angle is half the hypotenuse, and the side opposite the 60-degree angle is the hypotenuse divided by the square root of 3.
Given that the hypotenuse is 4, we can find the length of the sides as follows:
Side opposite the 30-degree angle = 4/2 = 2
Side opposite the 60-degree angle = 4/sqrt(3) = (4 * sqrt(3))/3
Now, to find the perimeter of the triangle, we add up the lengths of all three sides:
Perimeter = 2 + (4 * sqrt(3))/3 + 4
Simplifying the expression, we get:
Perimeter = 6 + (4 * sqrt(3))/3
Therefore, the correct answer is: 6 + 2 sqrt 3.
To find the perimeter of a triangle, we need to add the lengths of all three sides. In this case, we are given the length of the hypotenuse, which is 4.
Since we know the angles of the triangle (30°, -60°, -90°), we can determine the lengths of the other two sides using trigonometric ratios.
In a 30°, -60°, -90° triangle, the sides are in a ratio of 1:2:√3. This means that the length of the shorter leg (opposite the 30° angle) is (1/2) multiplied by the hypotenuse, and the length of the longer leg (opposite the -60° angle) is √3 multiplied by the shorter leg.
Let's calculate these lengths:
Shorter leg = (1/2) * 4 = 2
Longer leg = √3 * 2 = 2√3
Now we can find the perimeter:
Perimeter = Hypotenuse + Shorter leg + Longer leg
= 4 + 2 + 2√3
= 6 + 2√3
Therefore, the correct answer is 6 + 2√3.