the base of an isosceles triangle measures 108 feet and forms an angle of 55 degrees with one equal sides. find the perimeter of the triangle
Since there are two angles of 55°, the vertex angle is 70°. So, each of the two matching sides has length
54/cos55°
Now you can add up the sides to get the perimeter.
2(84.00909)+108=276.01818
To find the perimeter of the isosceles triangle, we need to know the lengths of the two equal sides. However, we are only given the length of the base and the measure of the angle between the base and one of the equal sides.
Here's how we can solve it step by step:
1. Draw the isosceles triangle:
- Start by drawing a line segment of length 108 feet. This will be the base of the triangle.
- From one of the endpoints of the base, draw another line segment that forms an angle of 55 degrees with the base. This line segment represents one of the equal sides of the triangle.
- Connect the other endpoint of the base to the endpoint of the line segment you just drew. This will complete the isosceles triangle.
2. Use the Law of Sines to find the length of the equal sides:
- The Law of Sines states that for any triangle: a/sin(A) = b/sin(B) = c/sin(C), where a, b, and c are the lengths of the sides, and A, B, and C are the opposite angles, respectively.
- In our case, we know the angle A (55 degrees) and the length of the base (108 feet).
- Let's call the length of the equal sides "x." We can label one of the equal angles as B, which makes the other equal angle C.
- Now, we have the equation: 108/sin(55°) = x/sin(B).
- Solving for sin(B), we get sin(B) = x / (108 / sin(55°)).
3. Calculate sin(B):
- Using a calculator or trigonometric table, find the value of sin(55°).
- Substitute this value into the equation sin(B) = x / (108 / sin(55°)) and solve for sin(B).
4. Find the length of the equal sides (x):
- Multiply sin(B) by (108 / sin(55°)) to find the length of the equal sides (x). Round the result to the appropriate number of decimal places.
5. Calculate the perimeter:
- Since the triangle is isosceles, it has two equal sides. Multiply the length of one equal side (x) by 2.
- Add the length of the base (108 feet) to the product from the previous step to get the perimeter of the triangle.
By following these steps, you can find the perimeter of the isosceles triangle.