The ratio of the sides is constant for similar triangles.
The angle between the edges of a polygon is 90 degrees.
In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
The side opposite the largest angle is the longest.
Right triangles.
Equilateral triangles.
Isosceles triangles.
Scalene triangles.
5.
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4.
a² + b² = c², where 'c' is the hypotenuse and 'a' and 'b' are the other two sides.
a + b = c, where 'c' is the circumference of a right triangle.
a² + b² = 2c, where 'c' is the altitude.
a² − b² = c², where 'c' is the height.
It's used for adding economic values in stock markets.
It is primarily used in biological experiments.
It can be used in construction, navigation, and any field requiring distance measurement.
It's used for solving chemical equations.
Three positive integers a, b, and c, such that a² + b² = c².
Three numbers that form a ratio of 1:2:3.
A set of numbers expressing angles in a right triangle.
Three decimal numbers whose product is zero.
(3, 4, 5).
(1, 2, 2).
(5, 12, 13).
(4, 7, 9).
Pythagoras, though some ancient civilizations knew it before him.
Euclid, the father of geometry.
Archimedes, known for his work in physics.
Aristotle, the philosopher.
By calculating the distance between two points using the distance formula derived from the theorem.
By determining the slope of a line in a coordinate plane.
By finding the midpoint between two points on a plane.
By calculating the area of any polygon.
It helps establish the basic trigonometric identities and relations between sine, cosine, and tangent.
It defines the tangent as the ratio of opposite to adjacent sides.
It provides the main formula for calculating sine.
It is used only to define cotangent relationships.
Yes, it can be extended to calculate distances in 3D space using the three dimensions.
No, it's exclusive to 2D geometry.
Only for calculating the volume of a cube.
Yes, but only for measuring angles, not distances.
If a² + b² = c² for a triangle's sides, then the triangle is a right triangle.
If a² + b² > c², then the triangle is obtuse.
If a² + b² < c², then the triangle is acute.
The opposite angles are equal.
Yes, like the law of cosines in any triangle and Fermat's Last Theorem for non-whole number powers.
No, there are no known generalizations beyond triangles.
Yes, it extends directly to arithmetic sequences.
Yes, to every shape that can create a square.
Using geometric proofs, such as rearranging squares on the triangle's sides.
With a bar chart of angles.
With a histogram of side lengths.
By drawing parallel lines across the triangle.
It illustrates the relationship between numbers and geometry, marking a fundamental principle in mathematics.
It was the first mathematical theory to involve calculus.
It was initially developed to solve problems in astronomy.
It signified the discovery of complex numbers in mathematics.