Find the length of |FG| - Leaving Cert Mathematics - Question (ii) - 2013

To find the length of |BD|, we apply the Cosine Rule:

1. The Cosine Rule states that: $c^2 = a^2 + b^2 - 2ab \cos(C)$ where $a$, $b$ are the lengths of sides adjacent to angle $C$, and $c$ is the side opposite.
2. Substituting the lengths, we have: $|BD|^2 = 8^2 + 9^2 - 2(8)(9)\cos(60^\circ)$ $= 64 + 81 - 72$ $= 73$
3. Hence, we find: $|BD| = \sqrt{73} \approx 8.544 m$.

To find the distance from point O to point B, we will employ the following:

1. From the relation given: $|OD| = \frac{x + 844}{|OB|} = 1:25$
2. Rearranging gives: $x + 844 = 125x$ which leads to: $0.025x + 844 = x$
3. Solve for $x$: $x = 341.76 m$.

To verify whether the plan meets the conditions, we analyze triangle relationships:

1. We know: $\angle GFH = \angle \alpha = \angle CBD$
2. Using the Law of Sines in triangle $GBD$: $\sin \alpha = \frac{9}{\sin 60^\circ} = \frac{9}{8544}$
3. And for triangle $GFH$: $\sin \alpha = \frac{h}{12.5} = \frac{9 \sin 60^\circ}{8544}$
4. From simplification, we can find: $h = 12.5 \times \frac{9 \sin 60^\circ}{8544} \approx 1.14 < 116$
5. Therefore, yes, the plan meets the condition.