Adelina is creating a piece of art that is in the shape of a parallelogram. its dimensions are 35 inches by 28 inches and one angle is 80 degrees. find the lengths of both diagonals.
To find the lengths of both diagonals, we can use the properties of a parallelogram.
First, let's label the given information:
Length of one side: 35 inches
Length of adjacent side: 28 inches
Measure of one angle: 80 degrees
To find the lengths of both diagonals, we need to use trigonometric ratios. Let's start with one diagonal, which we'll label as "d1."
1. Finding the length of diagonal d1:
We can divide the parallelogram into two congruent triangles. Let's label one of these triangles as triangle ABC.
Considering triangle ABC:
- The length of side AB is 35 inches.
- The length of side BC is 28 inches.
- We know one angle, angle B, which measures 80 degrees.
Using the Law of Cosines, we can determine the length of side AC (which is the same as the length of diagonal d1):
AC² = AB² + BC² - 2 * AB * BC * cos(angle B)
Plugging in the values we know:
AC² = 35² + 28² - 2 * 35 * 28 * cos(80)
Calculating this, we find:
AC ≈ 54.41 inches
So, the length of diagonal d1 is approximately 54.41 inches.
2. Finding the length of the other diagonal, d2:
Since opposite sides of a parallelogram are congruent, diagonal d2 has the same length as d1.
Therefore, the length of diagonal d2 is also approximately 54.41 inches.
To summarize:
Length of diagonal d1 is approximately 54.41 inches.
Length of diagonal d2 is also approximately 54.41 inches.
To find the lengths of both diagonals of the parallelogram, we can use the properties of the parallelogram and trigonometry.
The first step is to draw the parallelogram with the given dimensions. Since one angle is 80 degrees, we can label the sides and angles of the parallelogram as follows:
_____________________________
| |
| |
35 | |
________|______________________________|
28
Let's label the vertices of the parallelogram as A, B, C, and D:
A_______B
/ /
/ /
/ /
C_______/D
We know that opposite sides of a parallelogram are equal in length. So we can label the sides accordingly:
A_______B
/ 35 /
/ /
/ /
C_______/D
Now, we need to find the lengths of the diagonals AC and BD.
To find the length of diagonal AC, we can use the Law of Cosines. The Law of Cosines states that for any triangle with sides a, b, and c, and angle C opposite side c, we have the formula:
c^2 = a^2 + b^2 - 2ab*cos(C)
In triangle ABC, we have:
- Side a = 35 inches (opposite angle B)
- Side b = 28 inches (opposite angle C)
- Angle C = 80 degrees
Using the Law of Cosines, we can solve for the length of AC:
AC^2 = 35^2 + 28^2 - 2(35)(28)*cos(80)
Now we can calculate AC:
AC = sqrt(35^2 + 28^2 - 2(35)(28)*cos(80))
To find the length of diagonal BD, we can use the same approach. In triangle BCD, we have:
- Side a = 35 inches (opposite angle D)
- Side b = 28 inches (opposite angle C)
- Angle C = 80 degrees
Using the Law of Cosines, we can solve for the length of BD:
BD^2 = 35^2 + 28^2 - 2(35)(28)*cos(80)
Now we can calculate BD:
BD = sqrt(35^2 + 28^2 - 2(35)(28)*cos(80))
By plugging in the values and performing the calculations, we can find the lengths of both diagonals.