find the perimeter of a 30-60-90 degree triangle whose shorter leg is 16 cm
A 30-60-90 triangle has sides 1, √3 and 2 for a perimeter of 3+√3
since the shorter side of your triangle is 16, the perimeter must be 16 times as large or 48 + 16√3
The shorter leg is adjacent to the 60° angle.
We know from trigonometry that
sin 60° = 1/2
tan 60° = √3
Therefore:
adjacent side, A = 16 cm
opposite side, O = 16 tan 60° = 16√3
hypotenuse, H = 16 /sin 60° = 32
The perimeter is the sum of the three sides, namely
perimeter = A+O+H=16(3+√3)
Well, the perimeter of the triangle is going to be the sum of all the sides. In a 30-60-90 triangle, the sides are in a ratio of 1:√3:2. Since the shorter leg is 16 cm, we can find the other sides by multiplying it by the appropriate ratios. The longer leg will be 16√3 cm, and the hypotenuse will be 32 cm.
So, the perimeter will be 16 cm + 16√3 cm + 32 cm. But hey, there's no need to be triangular about it! That simplifies to 48 cm + 16√3 cm, or approximately 48 cm + 27.71 cm. Adding those up, we get a perimeter of approximately 75.71 cm. Ta-da!
To find the perimeter of a 30-60-90 degree triangle, we need to know the lengths of the sides. A 30-60-90 triangle is a special right triangle where the angles are 30 degrees, 60 degrees, and 90 degrees, and the sides are in a particular ratio.
In a 30-60-90 triangle, the ratio of the lengths of the sides is 1:√3:2. The shorter leg is the side opposite the 30-degree angle, the longer leg is opposite the 60-degree angle, and the hypotenuse is opposite the 90-degree angle.
In this case, the shorter leg is 16 cm. To find the lengths of the other sides, we can use the ratio of 1:√3:2.
Shorter leg = 1x
Longer leg = √3x
Hypotenuse = 2x
Now, we can solve for x by setting up an equation using the lengths of the sides:
1x = 16 cm
Dividing both sides of the equation by 1, we find that x = 16 cm.
Now we can find the lengths of the other sides:
Longer leg = √3x = √3 * 16 cm ≈ 27.71 cm
Hypotenuse = 2x = 2 * 16 cm = 32 cm
Finally, we can calculate the perimeter by adding the lengths of all three sides:
Perimeter = Shorter leg + Longer leg + Hypotenuse
Perimeter = 16 cm + 27.71 cm + 32 cm
Perimeter ≈ 75.71 cm
So, the perimeter of the 30-60-90 triangle with a shorter leg of 16 cm is approximately 75.71 cm.