In the given figure, $P, Q$, and $R$ are three points on a circle of radius 10 cm with $O$ as its center, $\overline{PQ} = \overline{RQ}$, and $\angle PQR = 45^\circ$. The figure is representative.
The area of the shaded region $PQRO$ is ______________ cm$^2$.
To find the area of the shaded region \(PQRO\), we follow these steps:
- Given that points \(P, Q,\) and \(R\) lie on a circle with radius 10 cm, and \( \overline{PQ} = \overline{RQ} \), \( \triangle PQR\) is isosceles.
- The angle \( \angle PQR = 45^\circ \).
- Since \(PQ = RQ\) and \(QO\) is a bisector, the angle \(\angle PQO = \angle RQO = \frac{45^\circ}{2} = 22.5^\circ\).
- The line segments \(PQ\) and \(RQ\) are chords of the circle.
- Using the property that the sum of angles in a triangle is \(180^\circ\), we have: \(\angle POQ = 180^\circ - 45^\circ = 135^\circ\\)
- To find the area of \( \triangle POQ \) and \( \triangle ROQ \), use the formula for the area of a triangle: \(\text{Area} = \frac{1}{2} \times r^2 \times \sin(\theta)\) where \(\theta\) is the angle between the lines.
- Calculate the area of each sector:
- The area of sector \(POQ = \frac{1}{2} \times 10^2 \times \sin(135^\circ) = 50\sin(135^\circ)\).
- Since \(\sin(135^\circ) = \sin(180^\circ - 135^\circ) = \sin(45^\circ) = \frac{\sqrt{2}}{2}\), the area of sector \(POQ\) becomes \(50 \times \frac{\sqrt{2}}{2} = 25\sqrt{2}\) cm\(^2\).
- The area of sector \(ROQ\) is equal to that of sector \(POQ\) due to symmetry: \(25\sqrt{2}\) cm\(^2\).
- The total area of \( \triangle PQRO \) is the area of both sectors: \(\text{Area} = 25\sqrt{2} + 25\sqrt{2} = 50\sqrt{2} \text{ cm}^2\).
Thus, the area of the shaded region \(PQRO\) is \(50\sqrt{2} \text{ cm}^2\).
Important Questions from Mensuration and Geometry
The city of Atlantis was crafted by the God of the seas, Poseidon. It was made of alternating concentric circular rings of land (shaded) and water (not shaded) as represented in the figure (not to scale). The radius of Inner Island was 2.5 stades (a unit of length used in ancient Greece). The water surrounding Inner Island was one stade wide (length AB). This was surrounded by two pairs of alternating rings of land and water. The first pair of land and water was two stades wide each (lengths BC and CD), and the outer pair is three stades wide each (lengths DE and EF).
The ratio of the surface area of the land to that of the water in the city of Atlantis is _________ (round off to two decimal places).
- A straight line $y = x - 1$ intersects a circle with center at $x = 1, y = 1$ and radius of magnitude 1 at two points. The length of the chord formed by this intersection is _______. (rounded off to three decimal places)
- The shell of a hollow spherical nanoparticle has a uniform thickness of 3 nanometers (nm). The outer radius of the nanoparticle is 5 nm. The ratio of the volume of the shell to the volume of the hollow core is ________
(Round off to one decimal place) - The volume of a sphere of diameter 1 unit is ______ than the volume of a cube of side 1 unit.
- A window is made up of a square portion and an equilateral triangle portion above it. The base of the triangular portion coincides with the upper side of the square. If the perimeter of the window is 6 m, the area of the window in $m^2$ is ______
