12 Topics of Geometry by Zeus Garcia

Foundation of Geometry (undefined terms, distance formula, midpoint formula)

4003297_1 — 2024-12-12 18:28:57 UTC

Undefined Terms: These are the building blocks of geometry that are not formally defined but are understood intuitively. The main undefined terms are:

Point:A location in space with no size or dimension.

Line: An infinite set of points extending in two directions with no thickness.

Plane: A flat surface that extends infinitely in all directions with no thickness.

Distance Formula: This formula calculates the straight-line distance between two points (x1,y1) and (x2,y2) in a coordinate plane:

d=(x2−x1)2+(y2−y1)2

Midpoint Formula: This formula finds the midpoint of a line segment joining two points (x1,y1) and (x2,y2)

M=(x1+x2,y1+y2)

Example: Using Distance Formula

Problem:Find the distance between the points A(2,3)A(2, 3)A(2,3) and B(5,7)B(5, 7)B(5,7).

Identify and Organize the Coordinates: Point A(2,3)A(2, 3)A(2,3): x1=2x_1 = 2x1=2, y1=3y_1 = 3y1=3.

Point B(5,7)B(5, 7)B(5,7): x2=5x_2 = 5x2=5, y2=7y_2 = 7y2=7.

Next substitute the values for Coordinates into distance Formula: d=(5−2)2+(7−3)2

Answer: The distance between A(2,3) and B(5,7)is 5 units.

Conditional Statements and Venn Diagrams

4003297_1 — 2024-12-13 18:22:02 UTC

A conditional statement is a logical statement that has two parts: a hypothesis and a conclusion, typically written in the form:If p, then q. There are three variants: Converse: Switch the hypothesis and conclusion: If q, then p.

Inverse: Negate both the hypothesis and conclusion: If not p, then not q. Contrapositive: Switch and negate the hypothesis and conclusion: If not q, then not p.

Venn Diagrams: are visual tools used to represent relationships between sets using overlapping circles. Formula: If p is true, then q must also be true.In a Venn diagram, p ⟹ q means the region representing p (the "if" part) is entirely contained within the region representing q (the "then" part). For example, If we express the implication p⟹q logically, it means that if someone likes chocolate, they must also like vanilla.

Parallel and Perpendicular lines

4003297_1 — 2024-12-13 18:25:23 UTC

Parallel lines are lines that never intersect and remain equidistant from each other, having identical slopes. Perpendicular lines intersect at a right angle, and their slopes are negative reciprocals of each other. For example, lines with slopes 2 and -1/2 are perpendicular.

Point-slope and Standard form

4003297_1 — 2024-12-13 18:30:56 UTC

The point slope form of a linear equation is written as y−y1=m(x−x1), where m is the slope of the line, and (x1,y1) is a point on the line. This form is useful when you know the slope and a specific point on the line.

The standard form of a linear equation is written as Ax+By=C, where A, B, and C are integers, and A is non-negative. This form is often used when you want to easily find intercepts or work with integer coefficients. For example, Given the slope m=2 and a point (3,4), the equation in point-slope form is:
y−4=2(x−3).

Parallelograms

4003297_1 — 2024-12-13 18:51:12 UTC

A parallelogram is a four-sided figure where opposite sides are both parallel and equal in length. The opposite angles are congruent, and adjacent angles are supplementary. The area is calculated using the formula Area=base×height. Parallelograms include special cases such as rectangles, rhombuses, and squares.

Triangle Congruence: SSS, SAS, ASA, AAS, HL

4003297_1 — 2024-12-13 19:03:57 UTC

Triangle congruence refers to the condition when two triangles are identical in size and shape, meaning all corresponding sides and angles are equal. There are several criteria to establish congruence: The SSS (Side-Side-Side) criterion states that if all three sides of one triangle are equal to the corresponding sides of another triangle, the triangles are congruent. The SAS (Side-Angle-Side) criterion means that if two sides and the included angle of one triangle are equal to the corresponding parts of another, the triangles are congruent. The ASA (Angle-Side-Angle) criterion asserts that if two angles and the included side of one triangle are equal to the corresponding parts of another triangle, the triangles are congruent. The AAS (Angle-Angle-Side) criterion applies when two angles and a non-included side are equal to the corresponding parts of another triangle. Lastly, the HL (Hypotenuse-Leg) criterion, which is specific to right triangles, states that if the hypotenuse and one leg of one right triangle are equal to the corresponding parts of another right triangle, the triangles are congruent. These criteria are used to prove that two triangles are congruent without needing to measure every side and angle. For example, for SSS (Side-Side-Side), if three sides of one triangle are equal in length to the corresponding sides of another triangle, the two triangles are congruent. For example, if triangle ABC has sides AB = 5, BC = 7, and CA = 8, and triangle DEF has sides DE = 5, EF = 7, and FD = 8, the triangles are congruent by SSS.

Rectangles

4003297_1 — 2024-12-13 19:24:18 UTC

A rectangle is a four-sided polygon, also known as a quadrilateral, where opposite sides are equal in length and all four angles are right angles, measuring 90 degrees. The opposite sides of a rectangle are parallel, and the diagonals are equal in length and bisect each other at the center.

The area of a rectangle is calculated by multiplying its length by its width. The perimeter is found by adding together the lengths of all four sides, which is equal to twice the sum of the length and the width.

The length of the diagonal of a rectangle can be determined using the Pythagorean theorem, where the diagonal is the square root of the sum of the squares of the length and width.

Rectangles are widely used in various areas such as architecture, design, and mathematics because of their simple, predictable properties.

Rhombi

4003297_1 — 2024-12-13 19:27:14 UTC

A rhombus is a type of quadrilateral where all four sides are of equal length. The opposite angles of a rhombus are equal, and its diagonals bisect each other at right angles (90 degrees). The diagonals also divide the rhombus into four congruent triangles.

The area of a rhombus can be calculated using the formula: Area = (diagonal 1 × diagonal 2) / 2.

The perimeter of a rhombus is found by multiplying the length of one side by four. Although a rhombus looks similar to a square, it does not necessarily have right angles.

Squares

4003297_1 — 2024-12-13 20:06:36 UTC

A square is a special type of rectangle and rhombus, characterized by having four equal sides and four right angles. In a square, each angle is 90 degrees, and the sides are all of equal length. The diagonals of a square are also equal in length, and they bisect each other at right angles, dividing the square into four equal right triangles. Additionally, the diagonals of a square act as axes of symmetry, meaning the square can be folded along these lines and still maintain its shape.

The area of a square is calculated by squaring the length of one of its sides, while the perimeter is found by multiplying the length of one side by four. For example, to find the area of the square, you square the length of one side:
Area = side × side = 5 × 5 = 25 square units.

Trapezoids

4003297_1 — 2024-12-13 20:10:01 UTC

A trapezoid is a quadrilateral with one pair of opposite sides that are parallel, called the bases, and the other pair of non-parallel sides called the legs. The height, or altitude, is the perpendicular distance between the two parallel sides. The area of a trapezoid is calculated using the formula: Area = 1/2 × (b₁ + b₂) × h, where b₁ and b₂ are the lengths of the two bases and h is the height.

The angles on the same side of a trapezoid are supplementary, meaning their sum is 180°. In an isosceles trapezoid, where the legs are of equal length, the trapezoid has reflectional symmetry. The diagonals of a trapezoid are not necessarily equal, but they intersect each other.

Midsegment of a trapezoid

4003297_1 — 2024-12-13 20:13:31 UTC

The midsegment of a trapezoid is the line segment that connects the midpoints of the non-parallel sides (legs). This segment is significant because it has special properties. The midsegment is parallel to both of the trapezoid’s bases and its length is the average of the lengths of the two bases.The formula for the length of the is: B=A+C/2

Kites

4003297_1 — 2024-12-13 20:15:56 UTC

A kite is a quadrilateral with two pairs of adjacent sides that are equal in length. Its diagonals intersect at right angles, with one diagonal bisecting the other. The longer diagonal is also a line of symmetry, dividing the kite into two congruent triangles. The area of a kite can be calculated using the formula: Area=1/2×d1×d2 d= diagonal length.

Citations

4003297_1 — 2024-12-13 20:22:00 UTC

1.https://images.app.goo.gl/9AEAKCqKpDqw3vYz9

2.https://images.app.goo.gl/gRy1WFk6L4HmqPH89

3.https://images.app.goo.gl/2yMGTP3M3QSrLXMJA

4.https://images.app.goo.gl/hTBg8XXRCw3fsm8P6

5.https://images.app.goo.gl/aRNyVThZMxHZLGQn6

6.https://images.app.goo.gl/yHAdHbwAJtHXj7Mm8

7.https://images.app.goo.gl/p5wr148vd1s6iEst9

8.https://images.app.goo.gl/XDxFFaf5DECwPYr49

9.https://images.app.goo.gl/sAV9j8aTjn9KcJX28

10.https://images.app.goo.gl/cXjGHvT6FDhx3Rqs8

11.https://images.app.goo.gl/9bDDYH9Kh9c7msePA

12.https://images.app.goo.gl/C5edPpKtvSRf8Rgj9