3.9.1 Pythagoras Theorem
3.9.1 Pythagoras Theorem
3.9.1 Pythagoras theorem
Pythagoras:
Pythagoras (c. 570 – c. 495 BC) was an ancient Greek philosopher and mathematician who is famous for the Pythagorean theorem, a fundamental result in geometry.
Pythagoras theorem:
The Pythagorean theorem states that in a right 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.
In mathematical terms, if a, b, and c are the lengths of the sides of a right triangle where c is the hypotenuse, then:
\( c^2=a^2+b^2 \)
The Pythagorean theorem is named after Pythagoras because he and his followers were the first to prove it and study its properties.
To use the Pythagorean theorem, you need to follow these steps:
Identify a right triangle.
A right triangle is a triangle that has one angle equal to \( 90 \) degrees.
Identify the two sides that form the right angle.
These two sides are called the legs of the triangle.
Identify the third side of the triangle, which is opposite the right angle.
This side is called the hypotenuse.
Label the lengths of the legs and the hypotenuse with variables, such as \( a, b \), and \( c \).
Write the Pythagorean theorem:
\( c^2=a^2+b^2 \)
Substitute the known lengths of the legs or the hypotenuse into the equation and solve for the unknown length.
Worked example:
Suppose you have a right triangle with legs of lengths \( 3 \) and \( 4 \) units.
To find the length of the hypotenuse, you can use the Pythagorean theorem as follows:
\( c^2=a^2+b^2 \)
\( c^2=3^2+4^2 \)
\( c^2=9+16 \)
\( c^2=25 \)
\( c=\sqrt{25} \)
\( c = 5 \)
Therefore, the length of the hypotenuse is \( 5 \) units.
In general, the Pythagorean theorem can be used to find the length of any side of a right triangle if the lengths of the other two sides are known.
This theorem is also used in various real-world applications, such as in construction, navigation, and astronomy, among others.