Question

find \\( \frac { g d } { m h } \\)
\\( \frac { g d } { m h } = \square \\) (type an integer or a simplified fraction.)

Explanation:

Step1: Determine if the triangles are similar

In \(\triangle DGF\), the third - angle \( \angle F=180^{\circ}-(69^{\circ}+39^{\circ}) = 72^{\circ}\). In \(\triangle HMK\), \(\angle M = 180^{\circ}-(75^{\circ}+30^{\circ})=75^{\circ}\). Wait, no, use the AA (Angle - Angle) similarity criterion.
In \(\triangle DGF\) and \(\triangle HMK\), \(\angle D = 39^{\circ}\), \(\angle H = 39^{\circ}\) and \(\angle F=72^{\circ}\), \(\angle K = 72^{\circ}\). So, \(\triangle DGF\sim\triangle HMK\) (by AA similarity: two pairs of corresponding angles are equal).

Step2: Use the property of similar triangles

For similar triangles \(\triangle DGF\) and \(\triangle HMK\), the ratios of corresponding sides are equal. That is \(\frac{GD}{MH}=\frac{GF}{MK}\)
We know \(GF = 18.2\) and \(MK = 8\)

Answer:

\(\frac{91}{40}\)