Find the lengths of the diagonals of a parallelogram with adjacent sides of 12cm and 13 cm if the angle between the sides is 50 degrees.
The shorter diagonal is what in terms of cm?
The longer diagonal is what in terms of cm?
Use cosine law
(shortone)^2 = 12^2 + 13^2 - 2(12)(13)cos50°
etc
for the longone, replace the end part with cos130°
To find the lengths of the diagonals of a parallelogram, we can use the law of cosines.
Let's label the adjacent sides of the parallelogram as a = 12cm and b = 13cm, and the angle between them as θ = 50 degrees.
The shorter diagonal, d1, can be found using the following formula:
d1^2 = a^2 + b^2 - 2ab*cos(θ)
Substituting the given values, we have:
d1^2 = (12)^2 + (13)^2 - 2(12)(13)*cos(50)
Calculating this expression:
d1^2 = 144 + 169 - 2(12)(13)*cos(50)
d1^2 = 313 - 312*cos(50)
d1^2 ≈ 313 - 245.90
d1^2 ≈ 67.10
Taking the square root of both sides, we find:
d1 ≈ √67.10
So the shorter diagonal is approximately equal to √67.10 cm.
To find the longer diagonal, d2, we can use the fact that opposite sides of a parallelogram are congruent:
d2 = d1 ≈ √67.10 cm.
Therefore, both the shorter and longer diagonals of the parallelogram are approximately √67.10 cm in length.
To find the lengths of the diagonals of a parallelogram, we can use the cosine rule. The cosine rule states that for a triangle with sides a, b, and c, and angle A opposite side a, the equation is given by:
c^2 = a^2 + b^2 - 2abcos(A)
In this case, we have a parallelogram with adjacent sides of 12cm and 13cm, and the angle between these sides is 50 degrees.
Let's calculate the shorter diagonal first:
Using the cosine rule, we have:
(diagonal)^2 = (side 1)^2 + (side 2)^2 - 2(side 1)(side 2)cos(angle between sides)
(diagonal)^2 = 12^2 + 13^2 - 2 * 12 * 13 * cos(50 degrees)
Simplifying this calculation, we get:
(diagonal)^2 = 144 + 169 - 312 * cos(50 degrees)
Now we can find the value of cos(50 degrees). Using a calculator, we get:
cos(50 degrees) ≈ 0.6428
Let's substitute the value of cos(50 degrees) back into the equation:
(diagonal)^2 = 144 + 169 - 312 * 0.6428
Simplifying further, we get:
(diagonal)^2 ≈ 144 + 169 - 200.6176
(diagonal)^2 ≈ 112.3824
Taking the square root of both sides, we find that the shorter diagonal is approximately:
diagonal ≈ √112.3824 ≈ 10.6 cm
Now, let's calculate the longer diagonal using the same process:
Using the cosine rule, we have:
(diagonal)^2 = (side 1)^2 + (side 2)^2 - 2(side 1)(side 2)cos(angle between sides)
(diagonal)^2 = 12^2 + 13^2 - 2 * 12 * 13 * cos(50 degrees)
(diagonal)^2 = 144 + 169 - 312 * cos(50 degrees)
Substituting the value of cos(50 degrees):
(diagonal)^2 = 144 + 169 - 312 * 0.6428
(diagonal)^2 ≈ 144 + 169 - 200.6176
(diagonal)^2 ≈ 112.3824
Taking the square root of both sides, we find that the longer diagonal is approximately:
diagonal ≈ √112.3824 ≈ 10.6 cm
Therefore, the lengths of both diagonals are approximately 10.6 cm.