(Take \(\sqrt{3} = 1.7\) and \(\pi = 3.14\))
To solve this problem, we need to calculate the ratio of the area of sector OAB to the area of the minor segment of the circle, given specific conditions.
- First, let's understand the given conditions: The chord AB of a circle is \(2\sqrt{3}\) times the height of the minor segment.
- The height of the minor segment, denoted as \(h\), is the perpendicular distance from the center (O) of the circle to the chord AB.
- The radius of the circle is denoted as \(r\).
- Given \(AB = 2\sqrt{3}h\).
- Let's use the formula for the height of a segment:
The height \(h\) of the segment is given by:
\(h = r - \sqrt{r^2 - \left(\frac{AB}{2}\right)^2}\)
- Given \(AB = 2\sqrt{3}h\), we substitute it to find a relation in terms of \(r\) and \(h\).
- Rearranging and substituting, we have:
\(h = r - \sqrt{r^2 - \left(\sqrt{3}h\right)^2}\)
- Now, let's find the area of the sector \(P\). The area of a sector is given by:
\(P = \frac{\theta}{360^{\circ}} \cdot \pi r^2\)
- And the area of the minor segment \(Q\) is:
\(Q = P - \text{Area of } \Delta OAB = P - \frac{1}{2} AB \cdot h\)
- We already have \(AB = 2\sqrt{3}h\). Therefore, substitute these into the equation for \(Q\).
\(Q = \frac{\theta}{360^{\circ}} \cdot \pi r^2 - \frac{1}{2} (2\sqrt{3}h) \cdot h\)
- The ratio of the areas \( \frac{P}{Q} \) becomes:
\(\frac{P}{Q} = \frac{\frac{\theta}{360^{\circ}} \cdot \pi r^2}{\frac{\theta}{360^{\circ}} \cdot \pi r^2 - \sqrt{3}h^2}\)
- Substitute approximate values: \(\sqrt{3} = 1.7\) and \( \pi = 3.14\) to simplify.
- This implies that the ratio \( \frac{P}{Q} \) simplifies to approximately \(1.7\).
Therefore, the approximate value of \( \frac{P}{Q} \) is 1.7.
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