To find the diagonal of a parallelogram, you connect to the opposite corner. Every parallelogram has two diagonals. To determine their specific lengths, you typically need to know the lengths of the sides of the parallelogram and the measure of at least one interior angle.
Understanding Parallelogram Diagonals
A diagonal in any quadrilateral, including a parallelogram, is a line segment that connects two non-adjacent vertices. As the reference mentions, a parallelogram always has two diagonals. These diagonals bisect each other, a key property of parallelograms, but this property doesn't directly give you the length of the diagonal itself.
To calculate the length of a diagonal, you can use geometric principles such as the Law of Cosines.
Methods for Calculating Diagonal Lengths
Finding the length of a parallelogram's diagonal requires using the side lengths and angles. Let the two adjacent side lengths of the parallelogram be denoted by 'a' and 'b', and let one of the interior angles between these sides be '$\theta$'. The other interior angle will be $180^\circ - \theta$.
There are two diagonals, let's call them $d_1$ and $d_2$.
Method 1: Using the Law of Cosines
This is the most common method when you know the lengths of two adjacent sides and the angle between them.
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Diagonal $d_1$: This diagonal connects vertices across the angle $\theta$. Using the Law of Cosines on the triangle formed by sides 'a', 'b', and diagonal $d_1$:
$d_1^2 = a^2 + b^2 - 2ab \cos(\theta)$
Therefore, $d_1 = \sqrt{a^2 + b^2 - 2ab \cos(\theta)}$ -
Diagonal $d_2$: This diagonal connects vertices across the angle $180^\circ - \theta$. Using the Law of Cosines on the triangle formed by sides 'a', 'b', and diagonal $d_2$:
$d_2^2 = a^2 + b^2 - 2ab \cos(180^\circ - \theta)$
Since $\cos(180^\circ - \theta) = -\cos(\theta)$, this simplifies to:
$d_2^2 = a^2 + b^2 + 2ab \cos(\theta)$
Therefore, $d_2 = \sqrt{a^2 + b^2 + 2ab \cos(\theta)}$
Summary Table:
| Diagonal | Formula (using side lengths 'a', 'b' and angle '$\theta$' between them) |
|---|---|
| $d_1$ | $\sqrt{a^2 + b^2 - 2ab \cos(\theta)}$ |
| $d_2$ | $\sqrt{a^2 + b^2 + 2ab \cos(\theta)}$ |
Note: $\theta$ is the angle between the sides 'a' and 'b'. One diagonal corresponds to the triangle with angle $\theta$, the other to the angle $180^\circ - \theta$. The formulas account for this difference in angles.
Method 2: Using the Parallelogram Law
The Parallelogram Law relates the lengths of the sides and the diagonals:
$d_1^2 + d_2^2 = 2(a^2 + b^2)$
This law states that the sum of the squares of the diagonals is equal to twice the sum of the squares of the sides. If you know the lengths of the two sides ($a$ and $b$) and the length of one diagonal ($d_1$), you can use this formula to find the length of the other diagonal ($d_2$).
$d_2^2 = 2(a^2 + b^2) - d_1^2$
$d_2 = \sqrt{2(a^2 + b^2) - d_1^2}$
Similarly, if you know $a$, $b$, and $d_2$, you can find $d_1$.
Practical Example
Suppose a parallelogram has sides of length 5 cm and 8 cm, and one interior angle is $60^\circ$.
Here, $a=5$, $b=8$, and $\theta = 60^\circ$. $\cos(60^\circ) = 0.5$.
The other angle is $180^\circ - 60^\circ = 120^\circ$. $\cos(120^\circ) = -0.5$.
Using Method 1:
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$d_1$ (across the $60^\circ$ angle):
$d_1 = \sqrt{5^2 + 8^2 - 2(5)(8)\cos(60^\circ)}$
$d_1 = \sqrt{25 + 64 - 80(0.5)}$
$d_1 = \sqrt{89 - 40}$
$d_1 = \sqrt{49} = 7$ cm -
$d_2$ (across the $120^\circ$ angle):
$d_2 = \sqrt{5^2 + 8^2 - 2(5)(8)\cos(120^\circ)}$
$d_2 = \sqrt{25 + 64 - 80(-0.5)}$
$d_2 = \sqrt{89 + 40}$
$d_2 = \sqrt{129} \approx 11.36$ cm
So the lengths of the diagonals are 7 cm and approximately 11.36 cm.
You could verify this using the Parallelogram Law:
$d_1^2 + d_2^2 = 7^2 + (\sqrt{129})^2 = 49 + 129 = 178$
$2(a^2 + b^2) = 2(5^2 + 8^2) = 2(25 + 64) = 2(89) = 178$
The values match, confirming the calculation.
