A diagonal of a parallelogram is 10 inches long and makes angles of 25 degrees and 43 degrees with the side. how long is the longest side?
a/sin A = b/sin B.
a/sin43 = 10/sin112, a = ?.
To find the length of the longest side of the parallelogram, we can use the law of cosines. Let's denote the length of the longest side as "x".
Step 1: Identify the given information:
- The length of a diagonal = 10 inches
- The angles formed between the diagonal and the side = 25 degrees and 43 degrees
Step 2: Apply the law of cosines:
According to the law of cosines, in a triangle with sides a, b, and c, and angle A opposite side a, the equation is:
c^2 = a^2 + b^2 - 2ab * cos(A)
In our case, consider the triangle formed by the diagonal and the two sides adjacent to the angles of 25 degrees and 43 degrees. Let's label the adjacent sides as "a" and "b" and the diagonal as "c".
Using the given information, we have the following equations:
- a^2 + b^2 - 2ab * cos(25) = 10^2
- a^2 + b^2 - 2ab * cos(43) = 10^2
Step 3: Solve the system of equations:
We have two equations with two unknowns (a and b). You can solve this system of equations using algebraic methods, such as substitution, elimination, or matrices.
Let's solve this using the substitution method. From the first equation, we can rewrite it as:
a^2 + b^2 - 2ab * cos(25) = 100
a^2 + b^2 = 100 + 2ab * cos(25)
Now, substitute this value of (a^2 + b^2) into the second equation:
(100 + 2ab * cos(25)) - 2ab * cos(43) = 100
2ab * cos(25) - 2ab * cos(43) = 0
ab * (cos(25) - cos(43)) = 0
Since ab cannot be equal to zero, we have:
cos(25) = cos(43)
Step 4: Solve for x (the longest side):
Now that we know cos(25) = cos(43), we can conclude that the parallelogram is a rectangle. In a rectangle, the diagonals are equal in length, therefore, the longest side of the parallelogram is also 10 inches.