Question

A rhombus is a quadrilateral with all sides equal in length. Recall that a rhombus is also a parallelogram. Find length AC and length BD in the rhombus shown here.

Solution

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Step 1
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Let point $P$ be the intersection between $\overline{AC}$ and $\overline{BD}$. By symmetry of the rhombus, we have the following equalities:

$$$\overline{BP}=\overline{PD}$$$

$$$\overline{AP}=\overline{PC}$$$

Equations (1) and (2) give rise to:

$$$2 \overline{AP} = \overline{AC}$$$

$$$2 \overline{BP} = \overline{BD}$$$

We also know from the givens of the problem that:

$$$\overline{AB}=\overline{BC}=\overline{CD}=\overline{DA}=18 \text{ in}$$$

By the symmetry of the rhombus, we also know the following facts:

$$$\overline{AC}\bot \overline{BD}$$$

$$$2 m\angle BAP = m \angle BAD$$$

$$$2m \angle ABP = m \angle ABC$$$

Equation (7) tells us that:

$$$m \angle BAP = 21 \text{\textdegree}$$$

The sine of this angle is the following:

$$$\sin(m \angle BAP) = \dfrac{\overline{BP}}{\overline{AB}}$$$

Therefore, we have

$$$\overline{BP} = \overline{AB}\sin(m \angle BAP)$$$

And by equation (4), we have:

$$$\overline{BD} = 2 \overline{AB}\sin(m \angle BAP)$$$

Substituting equation (7) and equation (5) gives:

$$$\overline{BD} = 2 (18)\sin(21 \text{\textdegree})$$$

$$$\boxed{\overline{BD} \approx 12.901 \text{ in}}$$$

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