a farmer has a triangular garden with two sides at right angles and the third side measuring 380 (feet long.)Find the perimeter of the garden if the angle between the shaded side and the longest side is 52(degrees)
If the other sides are a and b,
a/380 = sin 52°
b/380 = cos 52°
p = 380 + a + b
= 380(1 + sin52° + cos52°)
To find the perimeter of the garden, we need to calculate the lengths of the three sides and add them up.
Let's break down the problem:
1. We are given that one side of the garden measures 380 feet long.
2. The angle between the shaded side and the longest side is 52 degrees.
3. The garden has two sides at right angles, meaning they form a 90-degree angle.
To determine the lengths of the other two sides, we can use trigonometry. Specifically, we can use the trigonometric ratios - sine, cosine, and tangent.
In this case, we can use the sine function since we have the length of the opposite side (the shaded side) and the measure of the angle between them.
The sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse. In this case, the hypotenuse is the longest side of the triangle.
So, let's calculate the length of the shaded side:
sin(52 degrees) = Length of shaded side / Length of longest side (380 feet)
Length of shaded side = sin(52 degrees) * 380 feet
Now that we have the length of the shaded side, we can use the Pythagorean theorem to find the length of the third side.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
In this case, we have a right triangle with two sides at right angles (the two sides we need to find) and the longest side as the hypotenuse.
Let's call one of the unknown sides "a" and the other side "b."
Using the Pythagorean theorem, we have:
a^2 + b^2 = (Length of longest side)^2
Now, let's plug in the values:
a^2 + (sin(52 degrees) * 380 feet)^2 = (380 feet)^2
Solving for "a" will give us the length of the third side.
Once we have the lengths of all three sides, we can calculate the perimeter by adding them up.
Perimeter = Length of side 1 + Length of side 2 + Length of side 3
Therefore, we can sum up the process mentioned above to determine the perimeter of the triangular garden.