# data[].attributes.body
1. Point: A point is a location in space that has no size or dimension. It is represented by a dot and is named using a capital letter.

2. Line: A line is a straight path that extends infinitely in both directions. It is composed of an infinite number of points and has no thickness. It can be named using any two points on the line or a lowercase letter.

3. Plane: A plane is a flat, two-dimensional surface that extends infinitely in all directions. It is defined by three non-collinear points or by a capital letter.

4. Angle: An angle is formed when two rays share a common endpoint called the vertex. It is measured in degrees and can be classified as acute (less than 90 degrees), right (exactly 90 degrees), obtuse (greater than 90 degrees), straight (exactly 180 degrees), or reflex (greater than 180 degrees).

5. Circle: A circle is a set of points in a plane that are equidistant from a fixed center point. It is defined by its center and radius. The distance around the circle is called the circumference, and the distance across the circle passing through the center is called the diameter.

# data[].attributes.body
1. Acute Angle: An acute angle is an angle that measures less than 90 degrees. It is smaller than a right angle.

2. Right Angle: A right angle is an angle that measures exactly 90 degrees. It forms a perfect "L" shape.

3. Obtuse Angle: An obtuse angle is an angle that measures greater than 90 degrees but less than 180 degrees. It is larger than a right angle.

4. Straight Angle: A straight angle is an angle that measures exactly 180 degrees. It forms a straight line.

5. Reflex Angle: A reflex angle is an angle that measures greater than 180 degrees but less than 360 degrees. It is larger than a straight angle.

6. Full Rotation: A full rotation is an angle that measures exactly 360 degrees. It represents a complete revolution or a full circle.

# data[].attributes.body
Triangles can be classified based on their side lengths and their angles. Here are the different types of triangles:

Based on Side Lengths:

1. Equilateral Triangle: An equilateral triangle has three equal side lengths. All angles in an equilateral triangle are also equal, measuring 60 degrees each.

2. Isosceles Triangle: An isosceles triangle has two equal side lengths. The angles opposite the equal sides are also equal.

3. Scalene Triangle: A scalene triangle has no equal side lengths. All three sides have different lengths, and the angles can vary.

Based on Angles:

1. Acute Triangle: An acute triangle has three angles that are less than 90 degrees. All angles in an acute triangle are acute angles.

2. Right Triangle: A right triangle has one angle that measures exactly 90 degrees. The other two angles are acute angles.

3. Obtuse Triangle: An obtuse triangle has one angle that is greater than 90 degrees. The other two angles are acute angles.

It's important to note that a triangle can fall into multiple categories. For example, an equilateral triangle is also an acute triangle. Understanding these classifications helps in identifying and solving various geometric problems involving triangles.

# data[].attributes.body
Polygons are two-dimensional shapes with straight sides. They can be classified based on the number of sides they have. Here are the names and shapes of polygons from triangles to decagons:

1. Triangle: A triangle is a polygon with three sides.

2. Quadrilateral: A quadrilateral is a polygon with four sides. Examples include squares, rectangles, parallelograms, and trapezoids.

3. Pentagon: A pentagon is a polygon with five sides.

4. Hexagon: A hexagon is a polygon with six sides.

5. Heptagon: A heptagon is a polygon with seven sides.

6. Octagon: An octagon is a polygon with eight sides.

7. Nonagon: A nonagon is a polygon with nine sides.

8. Decagon: A decagon is a polygon with ten sides.

Each of these polygons has its own unique characteristics and properties. Understanding the names and shapes of different polygons helps in identifying and classifying geometric figures.

# data[].attributes.body
1. Center: The center of a circle is the point that is equidistant from all points on the circumference of the circle. It is often denoted by the letter "O".

2. Radius: The radius of a circle is the distance from the center to any point on the circumference. It is represented by the letter "r". The radius is half the length of the diameter.

3. Diameter: The diameter of a circle is a straight line segment that passes through the center and connects two points on the circumference. It is represented by the letter "d". The diameter is twice the length of the radius.

4. Circumference: The circumference of a circle is the distance around the outer boundary or perimeter of the circle. It is calculated using the formula C = 2πr, where "C" represents the circumference and "π" is a mathematical constant approximately equal to 3.14159.

5. Sectors: A sector is a region enclosed by two radii and the arc between them. It is similar to a slice of pie. The measure of a sector is given in terms of its central angle.

6. Segments: A segment is a region enclosed by a chord and the arc it intersects. It is like a piece of the circle, excluding the triangle formed by the chord.

7. Arcs: An arc is a curved part of the circumference of a circle. It is defined by two endpoints and the part of the circumference between them. Arcs can be minor arcs (less than 180 degrees), major arcs (greater than 180 degrees), or semicircles (exactly 180 degrees).

# data[].attributes.body
The Pythagorean Theorem is a fundamental principle in geometry that relates the lengths of the sides of a right triangle. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Mathematically, the Pythagorean Theorem can be expressed as:

a^2 + b^2 = c^2

Where:

- "a" and "b" represent the lengths of the two legs of the right triangle.

- "c" represents the length of the hypotenuse.

This theorem provides a powerful tool for solving various geometric problems involving right triangles. It is commonly used to find missing side lengths, determine if a triangle is a right triangle, or calculate distances in coordinate geometry.

The Pythagorean Theorem has applications in various fields, such as architecture, engineering, physics, and navigation. Understanding and applying this theorem is essential for working with right triangles and their properties.

# data[].attributes.body
1. Square:

- Area: A = side^2

- Perimeter: P = 4 * side

2. Rectangle:

- Area: A = length * width

- Perimeter: P = 2 * (length + width)

3. Triangle:

- Area: A = (base * height) / 2

- Perimeter: P = side1 + side2 + side3

4. Circle:

- Area: A = π * radius^2

- Circumference: C = 2 * π * radius (or C = π * diameter)

5. Parallelogram:

- Area: A = base * height

- Perimeter: P = 2 * (side1 + side2)

6. Trapezoid:

- Area: A = ((base1 + base2) * height) / 2

- Perimeter: P = side1 + side2 + side3 + side4

7. Ellipse:

- Area: A = π * major axis * minor axis

- Circumference (approximation): C ≈ π * (3 * (major axis + minor axis) - √((3 * major axis + minor axis) * (major axis + 3 * minor axis)))

8. Regular Polygon:

- Area: A = (perimeter * apothem) / 2

- Perimeter: P = number of sides * side length

# data[].attributes.body
1. Cube:

- Volume: V = side^3

- Surface Area: A = 6 * side^2

2. Rectangular Prism:

- Volume: V = length * width * height

- Surface Area: A = 2 * (length * width + width * height + height * length)

3. Cylinder:

- Volume: V = π * radius^2 * height

- Surface Area: A = 2 * π * radius * height + 2 * π * radius^2

4. Sphere:

- Volume: V = (4/3) * π * radius^3

- Surface Area: A = 4 * π * radius^2

5. Cone:

- Volume: V = (1/3) * π * radius^2 * height

- Surface Area: A = π * radius * (radius + slant height)

6. Cuboid (Rectangular Prism with unequal sides):

- Volume: V = length * width * height

- Surface Area: A = 2 * (length * width + width * height + height * length)

7. Pyramid:

- Volume: V = (1/3) * base area * height

- Surface Area: A = base area + (1/2) * perimeter * slant height

# data[].attributes.body
1. Similarity: Similarity refers to the relationship between two shapes that have the same shape but possibly different sizes. Two shapes are considered similar if their corresponding angles are equal and their corresponding sides are in proportion. In other words, the ratios of the corresponding side lengths of the shapes are equal.

2. Congruence: Congruence refers to the relationship between two shapes that have the same shape and size. Two shapes are considered congruent if all their corresponding angles are equal and all their corresponding side lengths are equal. Congruent shapes can be thought of as identical copies of each other.

# data[].attributes.body
Trigonometric ratios are mathematical functions that relate the angles of a right triangle to the ratios of its side lengths. The three primary trigonometric ratios are sine, cosine, and tangent. Here's a brief explanation of each ratio and how they relate to right triangles:

1. Sine (sin): The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. It is represented as sin(theta) or sin(angle). Mathematically, sin(theta) = opposite/hypotenuse.

2. Cosine (cos): The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. It is represented as cos(theta) or cos(angle). Mathematically, cos(theta) = adjacent/hypotenuse.

3. Tangent (tan): The tangent of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the adjacent side. It is represented as tan(theta) or tan(angle). Mathematically, tan(theta) = opposite/adjacent.

# data[].attributes.body
Coordinate geometry is a branch of mathematics that combines algebra and geometry. It involves plotting points on a graph using coordinates and using those coordinates to determine distances, midpoints, and other geometric properties. Here's a brief explanation of some key concepts in coordinate geometry:

1. Coordinate Plane: The coordinate plane is a two-dimensional plane formed by two perpendicular number lines called the x-axis and y-axis. It is divided into four quadrants.

2. Coordinates: Coordinates are pairs of numbers (x, y) that represent the position of a point on the coordinate plane. The x-coordinate represents the horizontal distance from the y-axis, and the y-coordinate represents the vertical distance from the x-axis.

3. Plotting Points: To plot a point on the coordinate plane, you locate its x-coordinate on the x-axis and its y-coordinate on the y-axis. The point is then represented by the intersection of the two lines.

4. Distance Formula: The distance between two points in the coordinate plane can be found using the distance formula. It is derived from the Pythagorean theorem and is given by the equation: d = sqrt((x2 - x1)^2 + (y2 - y1)^2), where (x1, y1) and (x2, y2) are the coordinates of the two points.

5. Midpoint Formula: The midpoint between two points in the coordinate plane can be found using the midpoint formula. It is given by the equation: (x, y) = ((x1 + x2)/2, (y1 + y2)/2), where (x1, y1) and (x2, y2) are the coordinates of the two points.

Coordinate geometry provides a powerful tool for analyzing and solving geometric problems using algebraic methods. It is widely used in various fields, including physics, engineering, computer graphics, and data analysis.

# data[].attributes.body
Transformations in geometry refer to the ways in which a shape can be manipulated or changed. Here are the four main types of transformations:

1. Translation: A translation, also known as a slide, involves moving a shape or object without changing its size, shape, or orientation. Every point of the shape is shifted by the same distance and in the same direction. Think of it as sliding the shape along a straight line.

2. Rotation: A rotation involves turning a shape or object around a fixed point called the center of rotation. The shape remains the same size and shape, but its orientation changes. The amount of rotation is measured in degrees.

3. Reflection: A reflection, also known as a flip, involves creating a mirror image of a shape or object. The shape is flipped over a line called the line of reflection. The distance of each point from the line of reflection remains the same, but the orientation is reversed.

4. Dilation: A dilation involves changing the size of a shape or object. It can either enlarge or reduce the size of the shape. The dilation is centered at a fixed point called the center of dilation, and the shape is scaled up or down by a certain factor.