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.
To solve this problem, we need to understand the geometry of the dodecagon and how to use its triangles to form squares.
- Each triangle in the dodecagon is formed by two radii of the circle and a side of the dodecagon. Since it is a regular dodecagon, it will have 12 such congruent isosceles triangles.
- The central angle of each triangle is calculated using: \(\frac{360^\circ}{12} = 30^\circ\).
- The side of each isosceles triangle (the dodecagon's side, \(d\)) can be found using the formula for the side of a polygon inscribed in a circle: \(d = 2r \sin(\frac{\pi}{12})\).
- We need to form squares with side length \(r\) using these triangles. To do this, consider how triangles can combine to form a square.
- Note that a square has a total interior angle sum of \(360^\circ\). Since each triangle has an apex angle of \(30^\circ\), and two base angles of \((180^\circ - 30^\circ)/2 = 75^\circ\), several triangles are required to sum up to \(360^\circ\).
- To form one square, we use 4 triangles, each contributing to a sum of \(4 \times 90^\circ = 360^\circ\) round angle at the center.
- Since 12 triangles are available, dividing by 4 gives us 3 complete squares.
Thus, the number of squares that can be formed is 3, and each square requires 4 triangles.
Therefore, the correct answer is: 3; 4
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.
- 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
Let $S$ be the portion of the plane $z = 2x + 2y - 100$ which lies inside the cylinder $x^2 + y^2 = 1$. If the surface area of $S$ is $\alpha\pi$, then the value of $\alpha$ is equal to ________.
