Find the perimeter and area of each figure. give your answers in the simplest radical form.a 30-60-90- degree triangle with hypotenuse length 28 centimeters.
X = 28cos30
X = 28*(sqrt3) / 2,
X = 14*sqrt(3).
Y = 28sin30 = 28 * 1/2 = 14cm.
P = 14*sqrt(3) + 14 + 28,
P = 14*sqrt(3) + 42.
A = X*Y/2,
A = (14sqrt(3)*14/2,
A = (14sqrt(3)*7,
A = 98sqrt(3).
In a 30-60-90 degree triangle, the side lengths are related by the following ratios:
- The length of the shorter leg is half the length of the hypotenuse.
- The length of the longer leg is √3 times the length of the shorter leg.
Given that the hypotenuse has a length of 28 centimeters, we can find the lengths of the other sides using these ratios.
1. Shorter leg:
The length of the shorter leg is half the length of the hypotenuse, so it is 28/2 = 14 centimeters.
2. Longer leg:
The length of the longer leg is √3 times the length of the shorter leg, so it is √3 * 14 = 14√3 centimeters.
3. Perimeter:
The perimeter of the triangle is the sum of the lengths of all three sides.
Perimeter = hypotenuse + shorter leg + longer leg
Perimeter = 28 + 14 + 14√3
Perimeter = 42 + 14√3 centimeters
4. Area:
The area of a 30-60-90 triangle can be found using the formula:
Area = (shorter leg * longer leg) / 2
Area = (14 * 14√3) / 2
Area = 196√3 / 2
Area = 98√3 square centimeters
So, the perimeter of the triangle is 42 + 14√3 centimeters, and the area is 98√3 square centimeters.
To find the perimeter and area of a 30-60-90 degree triangle, we need to know the lengths of the sides. In this case, we are given that the hypotenuse has a length of 28 centimeters.
In a 30-60-90 triangle, the sides are in a specific ratio. The ratio can be seen as follows:
- The side opposite the 30-degree angle has a length of x.
- The side opposite the 60-degree angle has a length of x√3.
- The hypotenuse has a length of 2x.
Since we are given the hypotenuse length, which is 28 centimeters, we can determine the length of the sides.
In this case:
- The side opposite the 30-degree angle is x = 28 / 2 = 14 centimeters.
- The side opposite the 60-degree angle is x√3 = 14√3 centimeters.
Now, we can calculate the perimeter and area of the triangle.
Perimeter:
The perimeter of any polygon is the sum of the lengths of all its sides. For a triangle, the perimeter can be calculated by adding the lengths of the three sides.
In this case, the perimeter of the triangle is:
14 + 14√3 + 28 = 42 + 14√3 centimeters.
Area:
The area of a triangle can be calculated using the formula:
Area = (base * height) / 2.
Since this is a right triangle, the base and height can be determined by taking the sides opposite the 90-degree angle.
In this case, the base and height are 14 centimeters and 14√3 centimeters, respectively.
Thus, the area of the triangle is:
(14 * 14√3) / 2
= 98√3 / 2
= 49√3 square centimeters.
Therefore, the perimeter of the triangle is 42 + 14√3 centimeters, and the area is 49√3 square centimeters.