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
In the given figure, PQRS is a square of side 2 cm and PLMN is a rectangle. The corner L of the rectangle is on the side QR. Side MN of the rectangle passes through the corner S of the square.
What is the area (in cm²) of the rectangle PLMN?
Note: The figure shown is representative.
A regular dodecagon (12-sided regular polygon) is inscribed in a circle of radius $r$ cm as shown in the figure. The side of the dodecagon is $d$ cm. All the triangles (numbered 1 to 12) in the figure are used to form squares of side $r$ cm and each numbered triangle is used only once to form a square.
The number of squares that can be formed and the number of triangles required to form each square, respectively, are:
Note: The figure shown is representative.
- Which one of the following options has the correct sequence of objects arranged in the increasing number of mirror lines (lines of symmetry)?
- A circle with center at $(x, y) = (0.5, 0)$ and radius $= 0.5$ intersects with another circle with center at $(x, y) = (1, 1)$ and radius $= 1$ at two points. One of the points of intersection $(x, y)$ is:
- During a half-moon phase, the Earth-Moon-Sun form a right triangle. If the Moon-Earth-Sun angle at this half-moon phase is measured to be $89.85^{\circ}$, the ratio of the Earth-Sun and Earth-Moon distances is closest to
