Complete the table by solving the parallelogram shown in the figure in the website link below. (The lengths of the diagonals are given by c and d. Round your answers to two decimal places.)
www.webassign.net/larpcalclim2/6-2-021.gif
a=22
b=39
c=
d=
θ= °
ϕ=122°
Taking angle ϕ, the diagonal opposite ϕ is
d^2 = 39^2 + 22^2 - 2*39*22*cos122° = 2914
d = 54
Now we know that ϕ and θ are supplementary, so
θ = 58°
Now use the law of cosines to find the other diagonal, if that's what you need. You can also find the altitude: 22 sinθ
hey oobleck thanks, but i still don't how to get c
I showed you how to get the long diagonal.
I also explained how to find the short diagonal and the altitude.
Surely you can't want anything else!
To solve the parallelogram and find the lengths of the diagonals (c and d), as well as the angles (θ and ϕ), we can use the properties and formulas of parallelograms. Let's break down the steps:
Step 1: Identify the given values:
- a = 22 (one side of the parallelogram)
- b = 39 (another side of the parallelogram)
- ϕ = 122° (an angle in the parallelogram)
Step 2: Find the missing values using the properties of parallelograms:
a) Calculating the diagonals:
We can use the properties of parallelograms to find the lengths of the diagonals.
In a parallelogram, the diagonals bisect each other, meaning they divide each other into equal halves.
Since the diagonals bisect each other, we can see that:
c = a/sin(ϕ) [Using the Law of Sines]
Substituting the given values, we have:
c = 22/sin(122°) = 22/ sin(122°) = 24.23 (rounding to two decimal places)
Similarly,
d = b / sin(ϕ) = 39 / sin(122°) = 43.26 (rounding to two decimal places)
So, c ≈ 24.23 and d ≈ 43.26.
b) Calculating the angles:
The opposite angles in a parallelogram are congruent. Since ϕ is an angle in the parallelogram, the opposite angle is also ϕ.
θ is the other angle in the parallelogram that complements ϕ. We can calculate θ using the following formula:
θ = 180° - ϕ
Substituting the given values, we have:
θ = 180° - 122° = 58°
Therefore, c ≈ 24.23, d ≈ 43.26, θ = 58°, and ϕ = 122°.
By following these steps, we can complete the table for the parallelogram.