Question

use the information given in the diagram of △ghi to find the missing measurements. round all answers to the nearest tenth.
m∠g = 51°
hi =
gh =
options for gh: 18.9, 23.3, 47.7
diagram: right triangle ghi with right angle at h, ∠i = 39°, side gi = 30.

Explanation:

Step1: Identify triangle type

△GHI is right-angled at H, so we can use trigonometric ratios (sine, cosine) to find the missing sides. Given \( m\angle I = 39^\circ \), \( m\angle G = 51^\circ \), and hypotenuse \( GI = 30 \).

Step2: Find HI (opposite to \( \angle G \))

Using sine: \( \sin(\angle G) = \frac{HI}{GI} \)
\( \sin(51^\circ) = \frac{HI}{30} \)
\( HI = 30 \times \sin(51^\circ) \)
\( \sin(51^\circ) \approx 0.7771 \)
\( HI \approx 30 \times 0.7771 \approx 23.3 \)

Step3: Find GH (adjacent to \( \angle G \))

Using cosine: \( \cos(\angle G) = \frac{GH}{GI} \)
\( \cos(51^\circ) = \frac{GH}{30} \)
\( GH = 30 \times \cos(51^\circ) \)
\( \cos(51^\circ) \approx 0.6293 \)
\( GH \approx 30 \times 0.6293 \approx 18.9 \)

Answer:

HI = 23.3
GH = 18.9