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

To find the diagonals of the parallelogram, use the law of cosines with the given sides and angles. Apply it separately for the smaller and larger diagonal using the angles 44° and 136° respectively, then take the square roots of the resulting values.

Step-by-step explanation:

To solve for the lengths of the diagonals of a parallelogram with sides 9 and 8 units, and one angle of 44°, we can use the law of cosines. The law of cosines states that for any triangle with sides a, b, and c, and angles A, B, C opposite those sides, the following equation applies: c² = a² + b² - 2ab*cos(C). Apply this formula to the two triangles formed by each diagonal of the parallelogram. For the smaller diagonal, apply the law of cosines using 44° as angle C, and for the larger diagonal, use 180° - 44° (the supplementary angle).

Step 1: Calculation for the smaller diagonal
Smaller Diagonal² = 9² + 8² - 2*9*8*cos(44°).
Step 2: Calculation for the larger diagonal
Larger Diagonal² = 9² + 8² - 2*9*8*cos(136°).

Once you have the squared lengths, take the square root of each to find the actual lengths of the diagonals.

Answer 2

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 = √(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