Geometry Basics

Study Notes

A point is a location in space, represented by a set of coordinates (x, y, z)
A line is a set of points that extend infinitely in two directions
A plane is a flat surface that extends infinitely in all directions
Intersections:
- Two lines intersect at a point
- A line and a plane intersect at a line
- Two planes intersect at a line

Angle: formed by two rays sharing a common endpoint (vertex)
Types of angles:
- Acute (< 90°)
- Right (= 90°)
- Obtuse (> 90°)
- Straight (= 180°)
Angle measurement:
- Degrees (°)
- Radians (rad)

Congruent shapes: same size and shape
Similar shapes: same shape, but not necessarily same size
Properties:
- Perimeter: distance around a shape
- Area: size of a shape's interior
- Volume: amount of space inside a 3D shape

Conversion between units:
- Length: 1 m = 100 cm = 1000 mm
- Capacity: 1 L = 1000 mL
- Mass: 1 kg = 1000 g
Conversion factors:
- 1 inch = 2.54 cm
- 1 foot = 12 inches

Precision: how close measurements are to each other
Accuracy: how close a measurement is to the true value
Rounding and estimating:
- Rounding: to a specific number of decimal places
- Estimating: approximating a value to a rough estimate

A point is a location in space, represented by a set of coordinates (x, y, z).
A line is a set of points that extend infinitely in two directions.
A plane is a flat surface that extends infinitely in all directions.
Intersections between geometric objects:
- Two lines intersect at a single point.
- A line and a plane intersect at a line.
- Two planes intersect at a line.

An angle is formed by two rays sharing a common endpoint (vertex).
Types of angles:
- Acute angles are less than 90°.
- Right angles are equal to 90°.
- Obtuse angles are greater than 90°.
- Straight angles are equal to 180°.
Angle measurement units:
- Degrees (°).
- Radians (rad).

Congruent shapes have the same size and shape.
Similar shapes have the same shape, but not necessarily the same size.
Properties of shapes:
- Perimeter is the distance around a shape.
- Area is the size of a shape's interior.
- Volume is the amount of space inside a 3D shape.

Conversion between units:
- 1 meter equals 100 centimeters or 1000 millimeters.
- 1 liter equals 1000 milliliters.
- 1 kilogram equals 1000 grams.
Conversion factors:
- 1 inch equals 2.54 centimeters.
- 1 foot equals 12 inches.

Precision refers to how close measurements are to each other.
Accuracy refers to how close a measurement is to the true value.
Rounding and estimating:
- Rounding involves rounding to a specific number of decimal places.
- Estimating involves approximating a value to a rough estimate.

An exponential function is a function of the form f(x) = a^x, where a is a positive real number and x is a real number.
The domain of an exponential function is all real numbers.
The range of an exponential function is all positive real numbers.
Exponential functions are always increasing or always decreasing, but never constant.
The graph of an exponential function has a horizontal asymptote at y = 0.
The graph of an exponential function increases rapidly as x increases.
The graph of an exponential function can be shifted vertically or horizontally by adding or multiplying by a constant.
Exponential functions can model population growth over time.
Exponential functions can model compound interest in finance.
Exponential functions can model chemical reactions and decay.

A polynomial function is a function of the form f(x) = a_n*x^n + a_(n-1)*x^(n-1) +...+ a_1*x + a_0, where a_n is a non-zero real number and n is a non-negative integer.
The degree of a polynomial function is the highest power of x.
The domain of a polynomial function is all real numbers.
The range of a polynomial function depends on the degree and coefficients of the polynomial.
The graph of a polynomial function can have multiple local maxima and minima.
The graph of a polynomial function can have vertical asymptotes.
The graph of a polynomial function can be shifted vertically or horizontally by adding or multiplying by a constant.
Monomials are polynomials with one term.
Binomials are polynomials with two terms.
Trinomials are polynomials with three terms.
Polynomial functions can model the trajectory of a projectile under gravity.
Polynomial functions can model the cost of producing a certain quantity of goods.
Polynomial functions can model the behavior of electrical circuits.