Pythagoras' Theorem
Definition:
Pythagoras' Theorem is one of the fundamental theorems in geometry, concerning right-angled
triangles. It states that, in a right-angled triangle, the square of the length of the hypotenuse (the
side opposite the right angle) is equal to the sum of the squares of the lengths of the other two
sides (the legs).
Statement of the Theorem:
In a right-angled triangle with legs of length and , and the hypotenuse of length , the theorem is
expressed as:
c^2 = a^2 + b^2
is the length of the hypotenuse.
and are the lengths of the legs.
Proof of the Theorem (Intuitive Explanation):
Imagine a square with side length containing a right-angled triangle. By constructing squares
on the sides of the triangle and calculating the areas, you'll discover that the area of the square
on the hypotenuse is equal to the sum of the areas of the squares on the legs.
This explanation can be further illustrated with diagrams showing triangles inside a square and
the areas of the squares built upon the sides of the triangle.
Properties of Pythagoras' Theorem:
Pythagorean Theorem's Converse: If, in a triangle, the relation holds true, then the triangle is
right-angled.
Relation to Right-Angled Triangles: The Pythagorean Theorem is a relationship that can be
used to determine the characteristics of right-angled triangles.
Generalization:
Definition:
Pythagoras' Theorem is one of the fundamental theorems in geometry, concerning right-angled
triangles. It states that, in a right-angled triangle, the square of the length of the hypotenuse (the
side opposite the right angle) is equal to the sum of the squares of the lengths of the other two
sides (the legs).
Statement of the Theorem:
In a right-angled triangle with legs of length and , and the hypotenuse of length , the theorem is
expressed as:
c^2 = a^2 + b^2
is the length of the hypotenuse.
and are the lengths of the legs.
Proof of the Theorem (Intuitive Explanation):
Imagine a square with side length containing a right-angled triangle. By constructing squares
on the sides of the triangle and calculating the areas, you'll discover that the area of the square
on the hypotenuse is equal to the sum of the areas of the squares on the legs.
This explanation can be further illustrated with diagrams showing triangles inside a square and
the areas of the squares built upon the sides of the triangle.
Properties of Pythagoras' Theorem:
Pythagorean Theorem's Converse: If, in a triangle, the relation holds true, then the triangle is
right-angled.
Relation to Right-Angled Triangles: The Pythagorean Theorem is a relationship that can be
used to determine the characteristics of right-angled triangles.
Generalization:
