Table of Contents
Meaning and Explanation of Pythagorean Theorem
The Pythagorean theorem states that in a right angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Right-Angled Triangle and Pythagorean Theorem
A right-angled triangle is a triangle that has one right angle. The Pythagorean theorem is a statement in mathematics that states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Right-Angled Triangle as a Combination of Three Squares
A right-angled triangle is a combination of three squares. The base of the triangle is a square, and the other two squares are located on the triangle’s hypotenuse. The length of the hypotenuse is the length of the triangle’s longest side.
Pythagorean Theorem Definition
The Pythagorean theorem is a geometric theorem that states that in a right angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
Geometrical Proof of Pythagorean Theorem
Theorem: In a right angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
Proof:
Let ABC be a right angled triangle with the right angle at C. Let the length of the hypotenuse be h, and the lengths of the other two sides be a and b.
We will show that:
h^2 = a^2 + b^2
We can use the Pythagorean theorem to write:
a^2 + b^2 = h^2
We can then subtract h^2 from both sides of the equation:
a^2 + b^2 – h^2 = 0
We can then divide both sides of the equation by a^2:
b^2 = h^2 – a^2
Pythagorean Theorem Examples
Example 1
A right triangle has the length of its longest side, or hypotenuse, as 12 feet. What are the lengths of the other two sides?
The other two sides have a length of 10 feet and 8 feet, respectively. This is because the Pythagorean theorem states that the sum of the squares of the two shorter sides is equal to the square of the length of the hypotenuse.
Application of Pythagorean Theorem
The Pythagorean theorem is one of the most famous geometric theorems. The theorem states that in a right angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. This theorem is named after the Greek mathematician Pythagoras, who is said to have discovered it. The theorem is often used in everyday life, for example when calculating the length of a piece of wood or the area of a square.
The Pythagorean theorem can be proved using a mathematical proof. The proof is based on the fact that in a right angled triangle, the angle at the right angle is 90 degrees. The proof starts by drawing a right angled triangle, with the angle at the right angle shown as a 90 degree angle.
Next, a line is drawn from the right angle to the longest side of the triangle, which is the hypotenuse. This line is shown as a dashed line in the diagram. A line is then drawn from the right angle to the other side of the triangle, which is the short side. This line is shown as a solid line in the diagram.
