Identifying Shapes by Increasing Lines of Symmetry
The question asks us to identify the sequence of geometric objects arranged in order of their mirror lines, also known as lines of symmetry, from least to greatest.
Understanding Lines of Symmetry
A line of symmetry is a line that divides a shape into two identical halves. When a shape is folded along its line of symmetry, both halves match exactly. We need to count these lines for each given shape.
Analyzing Mirror Lines for Each Object
Let's determine the number of lines of symmetry for each geometric object:
- Isosceles triangle: An isosceles triangle has exactly 1 line of symmetry. This line runs from the vertex between the two equal sides down to the midpoint of the base.
- Equilateral triangle: An equilateral triangle has 3 lines of symmetry. Each line goes from a vertex to the midpoint of the opposite side.
- Square: A square has 4 lines of symmetry. Two lines connect the midpoints of opposite sides, and the other two are the diagonals.
- Circle: A circle has an infinite number of lines of symmetry. Any line that passes through the center of the circle acts as a line of symmetry.
Ordering the Shapes
To find the correct sequence, we arrange the shapes based on the increasing number of their lines of symmetry:
- Isosceles triangle: 1 line of symmetry
- Equilateral triangle: 3 lines of symmetry
- Square: 4 lines of symmetry
- Circle: Infinitely many ($\infty$) lines of symmetry
The order of increasing lines of symmetry is therefore:
1 < 3 < 4 < $\infty$
This corresponds to the sequence:
Isosceles triangle; Equilateral triangle; Square; Circle
| Shape | Lines of Symmetry |
|---|---|
| Isosceles triangle | 1 |
| Equilateral triangle | 3 |
| Square | 4 |
| Circle | $\infty$ |
Final Sequence Check
Comparing this ordered sequence with the provided options, we find that "Isosceles triangle; Equilateral triangle; Square; Circle" represents the correct arrangement based on the increasing number of mirror lines.
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.
- 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 ________.
