Right Triangles: Pythagoras' Concept

Right Triangles

• A right triangle is a triangle with one right angle (90°).
• The two shorter sides of a right triangle are called legs, and the longest side is called the hypotenuse.

Pythagorean Theorem

• The Pythagorean Theorem states that for any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.
• The theorem can be represented mathematically as: a² + b² = c², where a and b are the legs and c is the hypotenuse.

Applications of the Pythagorean Theorem

• The theorem can be used to determine whether a triangle is a right triangle by checking if the equation a² + b² = c² is true.
• If the equation is true, the triangle is a right triangle; if not, it is not a right triangle.
• The theorem can be used to find the length of the hypotenuse or a leg of a right triangle, given the lengths of the other two sides.
• The theorem remains true if the lengths of the sides are multiplied by a constant factor.

Examples and Exercises

• In a triangle with sides 6, 7, and 11, the Pythagorean Theorem is not true, so it is not a right triangle.
• In a triangle with sides 3, 4, and 5, the Pythagorean Theorem is true, so it is a right triangle.
• The lengths of the perpendicular sides of a right-angled triangle are 12 cm and 16 cm; the length of the hypotenuse can be calculated using the Pythagorean Theorem.