Answer:
The length of the smaller diagonal is d_1 = 6.44 units
The length of the longer diagonal is d_2 = 15.77 units
Step-by-step explanation:
Given: A parallelogram has sides of lengths 9 and 8, and one angle is 44°.
We can use the cosine formula to find the measures of the diagonals.
The cosine formula is [tex]a^2 = b^2 + c^2 - 2bc (cos A)\\[/tex]
We can divide the parallelogram into two equal triangles.
Triangle 1
Which has the two sides measures 9 and 8 and including angle is 44°.
If we are given two sides and one angle, we can use the cosine formula and find the third side. Here third side is diagonal 1.
[tex]d_1^2 = 9^2 + 8^2 - 2*9*8 cos(44)\\d_1^2 = 81 + 64 - 103.58\\d_1 = 145 - 103.58\\d_1^2 = 41.42[/tex]
Taking square root on both sides, we get
[tex]d_1 = 6.44[/tex] [Rounded to the nearest hundredths place]
Triangle 2
It is a parallelogram, the opposite sides are equal. Therefore, the another triangle has equal measures but angle differs.
The adjacent angles in a parallelogram add upto 180°.
Therefore, the angle measure is 180 - 44 = 136°.
Now let's use the cosine formula and find the measure diagonal 2.
[tex]d_2 ^2 = 9^2 + 8^2 - 2*9*8*cos 136\\d_2^2 = 81 + 64 - (-103.58)\\d_2^2 = 145 + 103.58\\d_2^2 = 248.58[/tex]
Taking the square root on both sides, we get
[tex]d_2 = \sqrt{248.58} \\d_2 = 15.77[/tex] [Rounded to the nearest hundredths place]
So the length of the smaller diagonal is d_1 = 6.44 units
the length of the longer diagonal is d_2 = 15.77 units
