Geometry and Percentage Basics

Study Notes

Geometry

Definition: Study of shapes, sizes, relative positions of figures, and properties of space.
Basic Shapes:
- Triangle: Sum of interior angles = 180°. Types include equilateral, isosceles, and scalene.
- Square: All sides equal; interior angles = 90°; area = side².
- Rectangle: Opposite sides equal; area = length × width.
- Circle: A round shape; area = πr²; circumference = 2πr.
Key Theorems:
- Pythagorean Theorem: In a right triangle, a² + b² = c² (where c is the hypotenuse).
- Congruence: Two shapes are congruent if they have the same size and shape.
- Similarity: Two shapes are similar if they have the same shape but different sizes.

Calculating Percentage

Definition: A percentage is a fraction expressed as a part of 100.
Basic Formula:
- Percentage (%) = (Part / Whole) × 100
Finding the Part:
- Part = (Percentage / 100) × Whole
Finding the Whole:
- Whole = Part / (Percentage / 100)
Examples:
- If 25 out of 100 students pass an exam, the percentage of students passing is (25/100) × 100 = 25%.

Percentage Rate

Definition: The rate at which a quantity changes in terms of percentage.
Common Applications:
- Interest Rates: Used in finance to calculate growth on savings or payments on loans.
- Discounts: Calculation of price reductions on products.
Calculation:
- Percentage Rate = (Change in Value / Original Value) × 100
Example:
- If a product's price increases from 50to50 to 50to60, the percentage rate of increase is ((60−60 - 60−50) / $50) × 100 = 20%.

Geometry

Geometry involves the study of shapes, their sizes, placements, and properties of space.
Triangle classifications are based on side lengths: equilateral (all sides equal), isosceles (two sides equal), and scalene (all sides different). The sum of interior angles in a triangle is always 180°.
A square has equal sides and four right angles (90°), with area calculated as side².
Rectangles have opposite equal sides and the area is determined by multiplying length by width.
Circles are defined by their round shape, with area calculated using the formula πr² and the circumference using 2πr.
The Pythagorean Theorem is fundamental for right triangles, stating that the square of the hypotenuse (c) equals the sum of the squares of the other two sides (a² + b²).
Congruence refers to shapes that are identical in size and shape, while similarity means shapes have the same shape but may differ in size.

Calculating Percentage

A percentage is a numeric expression representing a fraction out of 100.
To calculate percentage: Percentage (%) = (Part / Whole) × 100.
To find the part of a whole given a percentage: Part = (Percentage / 100) × Whole.
To determine the whole given a part and its percentage: Whole = Part / (Percentage / 100).
For example, if 25 out of 100 students pass an exam, the calculation for passing percentage is (25/100) × 100, resulting in 25%.

Percentage Rate

Percentage rate measures how much a quantity increases or decreases in relation to its original value.
Common applications include calculating interest rates on savings or loans and determining discounts on products.
To compute the percentage rate of change: Percentage Rate = (Change in Value / Original Value) × 100.
For instance, if a product's price rises from 50to50 to 50to60, the percentage rate of increase is calculated as ((60−60 - 60−50) / $50) × 100, resulting in a 20% increase.

Western Asia

Located at the crossroads of Europe, Asia, and Africa, featuring diverse landscapes such as mountains, deserts, and fertile plains.
Comprises countries including Saudi Arabia, Iran, Iraq, Israel, Jordan, Lebanon, and Syria, among others.
Rich cultural diversity, with major religions including Islam, Christianity, and Judaism, and a variety of ethnic groups like Arabs, Persians, Kurds, and Turks.
Home to significant oil and gas reserves, especially in Gulf States; agriculture and tourism also play crucial roles in several economies.
Political landscape marked by ongoing conflicts, including the Israel-Palestine issue, the Syrian civil war, and tensions involving Iran and its neighbors.

Armenia

A landlocked nation situated in the South Caucasus region of Western Asia.
Historical significance as the first country to adopt Christianity as a state religion in 301 AD.
Notable for its rich cultural heritage, including the Armenian Genocide in 1915 during the Ottoman Empire.
Unique language and traditional arts, with a reputation for music and dance; famous for historical landmarks like monasteries and churches.
Functions as a republic and has complex relations with Azerbaijan, particularly concerning the Nagorno-Karabakh region.

Northern Cyprus

Geographically positioned on the northern part of Cyprus, along the Mediterranean Sea.
Declared independence in 1983; however, recognition is limited to Turkey.
History marked by conflict with the Republic of Cyprus, following the Turkish invasion in 1974.
Cultural influences from both Turkish and Greek traditions due to the island's historical context.
Economy heavily reliant on tourism and agriculture, facing challenges due to limited international recognition affecting economic growth.

Description

This quiz explores fundamental concepts in geometry including definitions and properties of basic shapes like triangles, squares, and circles. Additionally, it covers essential calculations related to percentages, including formulas for finding parts and wholes. Test your understanding of these mathematical principles!