Question

1. calculate the length of gh in the given diagram.

gc = 3.9 cm, (angle c = 90^circ), (angle cgd = 52^circ), (angle h = 90^circ), (angle hdg = 64^circ)

Explanation:

Step1: Identify right triangles

We have two right triangles: $\triangle GCD$ (right-angled at $C$) and $\triangle GHD$ (right-angled at $H$). First, solve for side $GD$ using $\triangle GCD$.

Step2: Calculate length of GD

Use cosine in $\triangle GCD$:
$\cos(52^\circ) = \frac{GC}{GD}$
Rearrange to solve for $GD$:
$GD = \frac{GC}{\cos(52^\circ)} = \frac{3.9}{\cos(52^\circ)}$
Calculate $\cos(52^\circ) \approx 0.6157$, so $GD \approx \frac{3.9}{0.6157} \approx 6.334$ cm

Step3: Calculate length of GH

Use sine in $\triangle GHD$:
$\sin(64^\circ) = \frac{GH}{GD}$
Rearrange to solve for $GH$:
$GH = GD \times \sin(64^\circ)$
Calculate $\sin(64^\circ) \approx 0.8988$, so $GH \approx 6.334 \times 0.8988 \approx 5.7$ cm

Answer:

$\boldsymbol{\approx 5.7}$ cm