Question

determine the answer to the following. * 11 points \\( \triangle mdh \\) is a right triangle where \\( m\angle dhm = 90^\circ \\), \\( m\angle dmh = 48^\circ \\), and \\( hm = 14 \\). 7. determine the length of \\( \overline{dh} \\) using a tangent ratio. show your work. if necessary, round your answer to the nearest thousandth. \\( \bigcirc 35 \\) \\( \bigcirc 48.888 \\) \\( \bigcirc 12 \\) \\( \bigcirc 63.458 \\) \\( \bigcirc 56.134 \\) \\( \bigcirc 61.928 \\) \\( \bigcirc 18 \\) \\( \bigcirc 40 \\) \\( \bigcirc 32.451 \\) \\( \bigcirc 53.130 \\) \\( \bigcirc 28.601 \\) \\( \bigcirc 45 \\) \\( \bigcirc 15.549 \\) \\( \bigcirc 14.665 \\)

Explanation:

Step1: Identify tangent ratio

For $\angle DMH = 48^\circ$, $\tan(\angle DMH) = \frac{\text{opposite}}{\text{adjacent}} = \frac{DH}{HM}$

Step2: Rearrange to solve for $DH$

$DH = HM \times \tan(\angle DMH)$
Substitute values: $HM=14$, $\angle DMH=48^\circ$
$DH = 14 \times \tan(48^\circ)$

Step3: Calculate the value

$\tan(48^\circ) \approx 1.110612515$
$DH \approx 14 \times 1.110612515 = 15.54857521$
Round to nearest thousandth: $DH \approx 15.549$

Answer:

15.549