A parallelogram has sides of lengths 7 and 4, and one angle is 55°. Find the lengths of the diagonals. (Round your answers to two decimal places. Enter your answers as a comma-separated list.)
To find the lengths of the diagonals of the parallelogram, we need to use the law of cosines.
1. Start by labeling the parallelogram. Let's say the sides of length 7 are opposite each other and the sides of length 4 are opposite each other.
A-------------B
/ \
/ \
D----------------C
Let AC represent the diagonal with length x, and let BD represent the other diagonal with length y.
2. Since the opposite sides of a parallelogram are equal in length, we have AD = BC and AB = CD.
3. We can use the law of cosines to find the lengths of the diagonals. Let's start with the diagonal AC.
In triangle ADC, we have:
AD^2 = AC^2 + CD^2 - 2 * AC * CD * cos(angle DAC)
Since AD = BC and AB = CD:
BC^2 = AC^2 + AB^2 - 2 * AC * AB * cos(angle DAC)
Substitute the values given:
4^2 = x^2 + 7^2 - 2 * x * 7 * cos(55°)
Simplify:
16 = x^2 + 49 - 14x * cos(55°)
Rearrange:
x^2 + 14x * cos(55°) - 33 = 0
4. Use the quadratic formula to solve for x:
x = [-14 * cos(55°) ± sqrt((14 * cos(55°))^2 - 4 * 1 * -33)] / (2 * 1)
Calculate the value inside the square root first:
(14 * cos(55°))^2 - 4 * -33 = 400
Then solve for x:
x = [-14 * cos(55°) ± sqrt(400)] / 2
x = [-14 * cos(55°) ± 20] / 2
x = -7 * cos(55°) ± 10
5. The lengths of the diagonals are usually positive, so we'll disregard the negative solution.
Hence, x = -7 * cos(55°) + 10.
6. Now let's find the length of the other diagonal BD.
We can use the same approach in triangle BCD:
CD^2 = BC^2 + BD^2 - 2 * BC * BD * cos(angle DCB)
Since AB = CD, AD = BC:
BC^2 = BD^2 + AD^2 - 2 * BD * AD * cos(angle DCB)
Substitute the values given:
7^2 = y^2 + 4^2 - 2 * y * 4 * cos(55°)
Simplify:
49 = y^2 + 16 - 8y * cos(55°)
Rearrange:
y^2 + 8y * cos(55°) - 33 = 0
7. Use the quadratic formula to solve for y:
y = [-8 * cos(55°) ± sqrt((8 * cos(55°))^2 - 4 * 1 * -33)] / (2 * 1)
Calculate the value inside the square root first:
(8 * cos(55°))^2 - 4 * -33 = 400
Then solve for y:
y = [-8 * cos(55°) ± sqrt(400)] / 2
y = [-8 * cos(55°) ± 20] / 2
y = -4 * cos(55°) ± 10
8. The lengths of the diagonals are usually positive, so we'll disregard the negative solution.
Hence, y = -4 * cos(55°) + 10.
9. Simplify the values of x and y:
x = -7 * cos(55°) + 10
y = -4 * cos(55°) + 10
Rounding to two decimal places, the lengths of the diagonals are approximately:
x = -7 * cos(55°) + 10 ≈ 7.42
y = -4 * cos(55°) + 10 ≈ 5.16
So, the lengths of the diagonals are approximately 7.42 and 5.16.
To find the lengths of the diagonals of a parallelogram, we can use the law of cosines. The law of cosines states that for any triangle with sides of length a, b, and c, and for any angle C opposite side c, the following equation holds:
c^2 = a^2 + b^2 - 2ab * cos(C)
In our case, we have a parallelogram with sides of lengths 7 and 4, and one angle of 55°. Let's label the sides of the parallelogram as follows:
A
--------------
/ \
/ D1 \
/ \
/ \
B S C
The diagonals of a parallelogram bisect each other, so we can draw line segments S and T connecting the midpoints of sides AB and CD, respectively. Let's label the lengths of these diagonals as D1 and D2, respectively.
Since opposite sides of a parallelogram are congruent, we know that AB = CD = 7 and AD = BC = 4.
We can see that triangle ABD is an isosceles triangle with base AB of length 7 and two congruent sides of length 4. The angle opposite side AB is 55°.
Using the law of cosines, we can calculate the length of the diagonal D1:
D1^2 = AB^2 + AD^2 - 2 * AB * AD * cos(angle ABD)
Substituting the known values:
D1^2 = 7^2 + 4^2 - 2 * 7 * 4 * cos(55°)
Calculate the value inside the cosine function:
cos(55°) ≈ 0.57357643635
D1^2 = 49 + 16 - 56 * 0.57357643635
D1 ≈ √(49 + 16 - 56 * 0.57357643635)
Calculate the approximate value of D1:
D1 ≈ √(49 + 16 - 56 * 0.57357643635) ≈ 7.09
Therefore, the length of the diagonal D1 is approximately 7.09.
Now, let's calculate the length of the other diagonal, D2:
Since the diagonals of a parallelogram bisect each other, we know that triangle ABD and triangle BCD are congruent.
Using the same calculation as before, we find that D2 ≈ 7.09 as well.
Thus, the lengths of the diagonals are approximately 7.09.
use the law of cosines. The short diagonal is
7^2+4^2-2*7*4cos55°
The longer diagonal lies opposite the supplementary angle, so it is
7^2+4^2+2*7*4cos55°