Question

A parallelogram has sides of lengths 9 and 8, and one angle is 44°. What is the length of the smaller diagonal? length = units What is the length of the longer diagonal? length = units

Answer 1

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 $a^2 = b^2 + c^2 - 2bc (cos A)$

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.

$d_1^2 = 9^2 + 8^2 - 2*9*8 cos(44)\\d_1^2 = 81 + 64 - 103.58\\d_1 = 145 - 103.58\\d_1^2 = 41.42$

Taking square root on both sides, we get

$d_1 = 6.44$ [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.

$d_2 ^2 = 9^2 + 8^2 - 2*9*8*cos 136\\d_2^2 = 81 + 64 - (-103.58)\\d_2^2 = 145 + 103.58\\d_2^2 = 248.58$

Taking the square root on both sides, we get

$d_2 = \sqrt{248.58} \\d_2 = 15.77$ [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