Key Math Concepts: Definitions and Theorems
A magnitude is anything that can be measured or counted.
Direct and Inverse Proportionality
Magnitudes directly proportional: If one magnitude is multiplied by a number, the other is divided by the same number automatically. For example, speed and time are inversely proportional; their product is constant.
Magnitudes inversely proportional: If one magnitude is multiplied or divided by a number, the other is divided or multiplied by that number, respectively.
Sequences
A sequence is an infinite set of numbers arranged in order, such that there is always a first term but no last term. Each number in a sequence is called a term, and the general term is denoted by n.
Arithmetic Progression
An arithmetic progression is a sequence in which each term, except the first, is obtained by adding a constant value (called the common difference) to the previous term.
Geometric Progression
A geometric progression is a sequence in which each term is obtained by multiplying the previous term by a fixed constant, called the common ratio.
Locus
A locus is a set of points that satisfy a given property. For example, a circle.
Circle
A circle is the locus of points in a plane that are equidistant from a fixed point called the center. The distance from any point on the circle to the center is called the radius.
Bisector of a Segment
The bisector of a segment is the locus of points that are equidistant from the endpoints of the segment.
Distance Between Two Points
The distance between two points is the length of the shortest path (a straight line) connecting them.
Angle Bisector
An angle bisector is the locus of points that are equidistant from both sides of the angle.
Medians of a Triangle
Medians are lines joining a vertex of a triangle to the midpoint of the opposite side.
Circumcenter of a Triangle
The circumcenter of a triangle is the point where the three perpendicular bisectors of the sides of the triangle intersect.
Height of a Triangle
The height of a triangle is the perpendicular distance from a vertex to the opposite side (or its extension).
Incenter of a Triangle
The incenter is the point where the angle bisectors of a triangle intersect. It is the center of the inscribed circle.
Pythagorean Theorem
Pythagorean Theorem: In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Inscribed Angle Theorem
The measure of an inscribed angle is half the measure of the central angle that subtends the same arc.
Sum of Interior Angles of a Convex Polygon
The sum of the interior angles of a convex polygon with n sides is 180° × (n – 2).
Circle Definition (Alternative)
A circle is the set of all points whose distance from the center is less than or equal to the radius of the circle.
Right Triangle Theorems
Theorem (Height): In a right triangle, the length of the altitude to the hypotenuse is the geometric mean of the lengths of the two segments it divides the hypotenuse into. (h2 = xy)
Theorem (Leg): In a right triangle, the length of each leg is the geometric mean of the length of the hypotenuse and the length of the projection of that leg onto the hypotenuse. (a2 = cx)
Polyhedron
A polyhedron is a three-dimensional solid bounded by polygons.
Prism
A prism is a polyhedron composed of two identical polygons (bases) and lateral faces that are parallelograms.
Thales’ Theorem
Thales’ Theorem: If a line is drawn parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally. The smaller triangle formed is similar to the original triangle.
Function
A function is a correspondence between two sets of numbers such that each element of the first set (the domain) is assigned to exactly one element of the second set (the range).
Solution of an Equation
A solution of an equation is a number that, when substituted for the variable, makes the equation true.