In a parallelogram, the lengths of the two adjacent sides are 12 cm and 14 cm respectively. If the length of one diagonal is 16 cm, find the length of the other diagonal.

In a parallelogram, the lengths of the two adjacent sides are 12 cm and 14 cm respectively. If the length of one diagonal is 16 cm, find the length of the other diagonal.
1). 14.6 cm
2). 20.6 cm
3). 18.3 cm
4). 25.2 cm

Let the parallelogram be $ABCD$ with sides $AB=12$ cm and $BC=14$ cm.
Let the diagonals be $AC$ and $BD$, where $AC=16$ cm.
The formula for the length of the diagonals in a parallelogram is given by:

$d_1^2+d_2^2=2(a^2+b^2)$

where
$a=12$ cm, $b=14$ cm, $d_1=16$ cm, and $d_2$ is the unknown diagonal.

Substituting the values:

${16}^2+d_2^2=2({12}^2+{14}^2)$

$256+d_2^2=2(144+196)$

$256+d_2^2=2(340)$

$256+d_2^2=680$

$d_2^2=680-256$

$d_2^2=424$

$d_2=\sqrt{424}$

$d_2\approx 20.6$ cm

Thus, the length of the other diagonal is:

$\boxed{{\displaystyle 20.6}}$ cm

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