The sides of a parallelogram are 15 ft and 17 ft. One angle is 40° while another angle is 140°. Find the lengths of the diagonals of the parallelogram (to the nearest tenth of a foot)
To find the lengths of the diagonals of a parallelogram, we can use the law of cosines. Let's label the sides and angles as follows:
Side 1: a = 15 ft
Side 2: b = 17 ft
Angle 1: α = 40°
Angle 2: β = 140°
The diagonals of a parallelogram bisect each other. Let's consider the diagonals as two line segments that divide the parallelogram into four triangles.
To find the lengths of the diagonals, we need to focus on one of the triangles. Let's consider the triangle formed by one side of the parallelogram, one diagonal, and half of the other diagonal.
In this right triangle, we have:
Side 1: a = 15 ft
Side 2: half of the diagonal
Angle: α = 40°
Now, we can use the law of cosines to find the length of the diagonal. The law of cosines states:
c^2 = a^2 + b^2 - 2 * a * b * cos(α)
Applying this formula to our triangle, we have:
diagonal^2 = a^2 + (0.5 * diagonal)^2 - 2 * a * 0.5 * diagonal * cos(α)
Simplifying this equation, we get:
diagonal^2 = a^2 + 0.25 * diagonal^2 - a * diagonal * cos(α)
Rearranging the equation, we have:
0.75 * diagonal^2 = a^2 - a * diagonal * cos(α)
Now, we can plug in the known values:
0.75 * diagonal^2 = 15^2 - 15 * diagonal * cos(40°)
To solve for the diagonal, isolate "diagonal":
0.75 * diagonal^2 + 15 * diagonal * cos(40°) - 15^2 = 0
This is a quadratic equation in terms of "diagonal." To solve it, we can use the quadratic formula:
diagonal = (-b ± √(b^2 - 4ac)) / (2a)
Applying this formula to our equation, we have:
diagonal = (-15 * cos(40°) ± √((15 * cos(40°))^2 - 4 * 0.75 * 15^2)) / (2 * 0.75)
Calculating this, we find two solutions for the diagonal. Since the diagonals bisect each other, we only need to consider one of the solutions.
Let's calculate the length of the diagonal:
diagonal = (-15 * cos(40°) + √((15 * cos(40°))^2 - 4 * 0.75 * 15^2)) / (2 * 0.75)
diagonal ≈ 25.1 ft (rounded to the nearest tenth of a foot)
Therefore, the length of one of the diagonals of the parallelogram is approximately 25.1 ft.
for the long diagonal
LD^2 = 15^2 + 17^2 - 2(15)(17)cos140°
solve for LD
let me know what your equation is for the short diagonal.