A parallelogram has sides with lengths of $14$ $14$ and $8$ $8$ . If the parallelogram's area is $84$ $84$ , what is the length of its longest diagonal?

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A parallelogram has sides with lengths of $14$ $14$ and $8$ $8$ . If the parallelogram's area is $84$ $84$ , what is the length of its longest diagonal?

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Answer 1 · 2016-05-16T20:05:24

The longest diagonal $= 20.203$ $=20.203$

Explanation:

The parallelogram with sides $a, b$ $a,b$ has an area computed as
$A = a \times b \times sin (θ) = 84$ $A = a times b times sin(theta) = 84$
Solving for $θ$ $theta$ we obtain $θ_{0} = 2.29353$ $theta_0 = 2.29353$ [rad]
Associating now an oriented segment to each side we get
$\to a = a {cos (θ_{0}), sin (θ_{0})}, \to b = b {1, 0}$ $vec a = a{cos(theta_0),sin(theta_0)}, vec b = b{1,0}$
The two diagonals are obtained as
${\to d}_{1} = \to a + \to b$ $vec d_1 = vec a + vec b$
${\to d}_{2} = \to a - \to b$ $vec d_2 = vec a - vec b$
Computing their length we get
$∥ ∥ ∥ {\to d}_{1} ∥ ∥ ∥ = 10.5753$ $norm(vec d_1) = 10.5753$
$∥ ∥ ∥ {\to d}_{2} ∥ ∥ ∥ = 20.203$ $norm(vec d_2) = 20.203$

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A parallelogram has sides with lengths of $14$ $14$ and $8$ $8$ . If the parallelogram's area is $84$ $84$ , what is the length of its longest diagonal?

1 Answer

Explanation:

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A parallelogram has sides with lengths of $14$ $14$ and $8$ $8$ . If the parallelogram's area is $84$ $84$ , what is the length of its longest diagonal?