Geometry

The problem involves cutting the corners of an equilateral triangle to form a regular convex hexagon. We need to find the ratio of the area of the regular convex hexagon to the area of the original equilateral triangle. Step 1: Understand the geometry of the problem.
When corners are cut off an equilateral triangle, the resulting shape is a regular convex hexagon. The key to solving this problem is recognizing that the area of the regular hexagon is proportional to the area of the equilateral triangle from which it is formed. Step 2: Calculate the area of the equilateral triangle.
The area of an equilateral triangle with side length is given by the formula: Step 3: Calculate the area of the regular hexagon.
The regular hexagon formed by cutting the corners of the equilateral triangle will have a side length that is a fraction of the side length of the equilateral triangle. After cutting off the corners, the remaining area is that of the regular hexagon. The area of the hexagon can be calculated using the formula for the area of a regular hexagon with side length : However, for this case, the area of the hexagon is proportional to the area of the original triangle, and the proportionality constant comes out to be . Step 4: Find the ratio of areas.
The ratio of the area of the regular hexagon to the area of the original equilateral triangle is: Thus, the ratio is , which corresponds to option (A). Final Answer: 2 : 3

We are given a square sheet with side 1 unit, and the triangle is formed by cutting along the diagonal. The next step involves revolving one of the triangles about its short edge, which will form a cone. Let's find the volume of this cone.
- The base radius of the cone is half of the side of the square, so .
- The height of the cone is the length of the other side of the triangle, which is also .
The formula for the volume of a cone is: Substituting the values of and : Thus, the volume of the cone is cubic units. Final Answer:

We are given a figure with a vertical dashed line labeled P-Q, which is intended to be the line of symmetry. The problem asks us to determine the least number of squares that must be added to the figure so that the line P-Q becomes the axis of symmetry for the entire arrangement.
Step 1: Analyze the initial figure.
The figure consists of several squares arranged around the line P-Q. To determine the number of squares that need to be added, we need to visualize what the figure would look like if it were symmetric along this line.
Step 2: Apply the concept of symmetry.
Symmetry in this case means that for each square on one side of the line P-Q, there must be a corresponding square on the opposite side. In this case, the figure is asymmetric along the line P-Q, which means that squares are missing on one side of the line.
Step 3: Determine the missing squares.
By observing the figure carefully, we can see that adding 6 more squares would complete the symmetry, making the entire shape symmetric about the line P-Q. Each new square will mirror the existing squares on the other side, ensuring that the figure is perfectly symmetrical.
Thus, the least number of squares to be added is 6.
Therefore, the correct answer is option (C).
Final Answer: 6

Step 1: Understand the situation.
The art gallery has an opaque boundary, and the security guard is positioned such that the entire inner space is visible within their 360° field of view. This means the security guard needs to be posted in locations where their view encompasses the entire shaded region of the gallery.

Step 2: Analyze the options.
- (A) P and Q: These two locations do not cover the entire shaded area of the gallery as the region behind point R is left out.
- (B) Q: This location only provides partial coverage, as it misses a portion of the gallery's inner space.
- (C) Q and S: Both Q and S locations together will cover the entire shaded region. Point Q covers the top portion, and point S covers the bottom, ensuring complete visibility.
- (D) R and S: These points miss certain areas in the middle of the gallery.

Step 3: Conclusion.
The correct answer is (C) Q and S, as these two locations together can watch over the entire inner space of the gallery.

The formula for the area of a quadrilateral with vertices at is: Substituting the coordinates of the points , we get: Thus, the area enclosed by the quadrilateral is .

The shortest distance between two parallel sides in a trapezium is the perpendicular distance between them. To find this, we can use the formula for the area of the trapezium and equate it to the sum of the areas of two triangles and a rectangle formed by the given dimensions.
First, calculate the area of the trapezium using the formula:
where and are the lengths of the parallel sides and is the perpendicular height (the shortest distance). We are given:
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- The total length of the non-parallel sides
Next, use the fact that the area of the trapezium can also be expressed as the area of the rectangle plus the two triangular areas formed by the slant sides. After solving the geometry and using the trapezium area formula, the shortest distance (height) is found to be:

Step 1: Understanding the structure of the grid
The grid has a total of 16 unit squares. One of these unit squares is a hole in the center. Therefore, we need to form squares without using the unit square at the center of the grid.

Step 2: Finding possible square sizes
- A square can be formed in any of the 15 remaining unit squares (excluding the center hole).
- A square can be formed by selecting four unit squares. In this case, the hole at the center prevents a square from being formed completely within the grid. Thus, we can form 5 such squares.
- A square can be formed by selecting a block of squares. The hole is in the interior, but it does not affect the construction of the square as the hole is on the edge, so we can form 1 such square.

Step 3: Summing the possible squares
Total number of squares that can be formed:
- 15 squares of size
- 5 squares of size
- 1 square of size
Thus, the maximum number of squares that can be formed without a "hole in the interior" is:

Step 1: Understanding the effects of increasing the width of a rectangle by 10%
The dimensions of the rectangle are (length) and (width), with . When the width is increased by 10%, the new width becomes , while the length remains the same.

Step 2: Analyzing the impact on each option

Option (A): Perimeter increases by 10%
The perimeter of a rectangle is given by the formula:

When the width increases by 10%, the new perimeter becomes:

This is not exactly a 10% increase. The increase in perimeter is not proportional to the increase in width. Therefore, this option is incorrect.

Option (B): Length of the diagonals increases by 10%
The diagonal of a rectangle is given by the Pythagorean theorem:

When the width increases by 10%, the new diagonal is:

This increase is not guaranteed to be exactly 10%. The length of the diagonal increases, but not necessarily by 10%. Therefore, this option is incorrect.

Option (C): Area increases by 10%
The area of a rectangle is:

After increasing the width by 10%, the new area is:

This shows that the area increases by exactly 10%. Therefore, this option is correct.

Option (D): The rectangle becomes a square
A rectangle becomes a square only if the length and width are equal. Since only the width is increased by 10% and , this option is incorrect.

Therefore, the correct answer is Option (C): Area increases by 10%.

In this problem, we are given a figure involving three geometric shapes: a rectangle, a triangle, and an ellipse. We need to determine the most appropriate values for , , and based on the dimensions provided in the figure.

The number is associated with the length of the rectangle's side. Since the side of the rectangle is labeled as 6, we can conclude that .
The number is associated with the height of the triangle. The height of the triangle is labeled as 5, so .
The number is associated with the length of the major axis of the ellipse. The length of the major axis is labeled as 3, so .

Thus, the correct values are , , and .

We are given a regular dodecagon inscribed in a circle, and we need to form squares using the triangles formed by connecting the center of the circle to the vertices of the dodecagon. There are 12 triangles in total, each corresponding to a side of the dodecagon. The number of squares that can be formed is 3, and each square requires 4 triangles. Hence, the correct number of squares and the number of triangles required to form each square are 3 and 4, respectively.