Fundamental Concepts of Algebra

Square Roots and Properties

Understanding square roots is essential for solving equations and working with geometric formulas. The square root of a negative number is not defined in the set of real numbers.

• Square Root Definition: For a positive number x, is the positive number whose square is x.

• Examples: , ,

• Properties:

• Not Defined: is not defined for real numbers.

Geometry: Lines and Angles

Parallel and Perpendicular Lines

Lines are fundamental objects in geometry. Parallel lines never intersect, while perpendicular lines intersect at a right angle.

• Parallel Lines: Notation:

• Perpendicular Lines: Notation:

• Angle: denotes the angle at point X.

Similar and Congruent Shapes

Shapes can be classified as similar or congruent based on their properties.

• Similar Shapes: Same shape, angles, and proportions.

• Congruent Shapes: Same shape, angles, proportions, and size.

Transformations

Transformations move or change shapes in the plane. Common transformations include translation, rotation, and reflection.

• Translation: Moves a shape without rotating or flipping it.

• Rotation: Turns a shape around a fixed point.

• Reflection: Flips a shape over a line (axis).

Circles

Area and Circumference

The area and circumference of a circle are fundamental geometric properties.

• Area: where r is the radius.

• Diameter:

• Circumference:

Annulus (Ring-Shaped Region)

An annulus is the region between two concentric circles.

• Area of Annulus: where R is the outer radius and r is the inner radius.

Sectors and Arcs

A sector is a region of a circle bounded by two radii and the arc between them. The arc is a portion of the circle's circumference.

• Area of Sector: where is the central angle in degrees.

• Length of Arc:

Central and Inscribed Angles

Central angles are formed at the center of the circle, while inscribed angles are formed on the circumference.

• Central Angle: The angle at the center; its measure equals the arc it intercepts.

• Inscribed Angle: The angle at the circumference; its measure is half the intercepted arc.

Triangles

Area and Perimeter of Triangles

The area and perimeter of triangles are calculated using their base and height or side lengths.

• Area: where b is the base and h is the height.

• Perimeter: where a, b, and c are the side lengths.

Equilateral Triangles

Equilateral triangles have all sides of equal length and all angles equal to 60°.

• Area: where s is the side length.

• Perimeter:

Rectangles and Squares

Area and Perimeter of Rectangles

Rectangles have opposite sides equal and four right angles.

• Area: where l is length and w is width.

• Perimeter:

Area and Perimeter of Squares

Squares have all sides equal and four right angles.

• Area:

• Perimeter:

Solids: Prisms, Cylinders, and Spheres

Rectangular Prisms

Rectangular prisms are three-dimensional solids with six rectangular faces.

• Surface Area:

• Volume:

Cylinders

Cylinders are solids with two parallel circular bases connected by a curved surface.

• Surface Area:

• Lateral Area:

• Volume:

Spheres

Spheres are perfectly round three-dimensional objects.

• Surface Area:

• Volume:

Right Triangles and the Pythagorean Theorem

Pythagorean Theorem

The Pythagorean Theorem relates the sides of a right triangle.

• Theorem: where c is the hypotenuse.

• Application: Used to find the length of a side when the other two are known.

Surface Area and Volume

Definitions

Surface area is the total area covering the surface of a 3D shape, while volume is the space contained within it.

• Surface Area: Measured in square units.

• Volume: Measured in cubic units.

Summary Table: Area and Volume Formulas