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Pythagoras Formula
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 other two sides. This theorem is represented by the equation: a^2 + b^2 = c^2.
Formula For Pythagoras 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 other two sides.
Use of Pythagorean Theorem Formula
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 other two sides. The theorem represented by the equation: a^2 + b^2 = c^2.
Pythagorean theorem can used to find the length of hypotenuse of a right angled triangle, given lengths of other two sides. To use theorem, squares of the lengths of other two sides added together to find the square of length of the hypotenuse.
Derivation of Pythagorean Theorem
Pythagorean theorem states that in a right angled triangle, sum of the squares of two shorter sides is equal to square of the length of long side.
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 other two sides. The theorem is named after the Greek mathematician Pythagoras.
There are many ways to prove the Pythagorean theorem. One way to prove the theorem is by using a method called “reduction to absurdity.” This method involves assuming that the theorem is not true, and then showing that this assumption leads to a contradiction.
For example, suppose that we have a right angled triangle with sides of length 3, 4, and 5. If Pythagorean theorem were not true, then the square of the length of the hypotenuse would not equal to the sum of the squares of the other two sides. This would mean that the length of the hypotenuse would be less than the sum of the other two sides.
However, we can use the Pythagorean theorem to show that this is not the case. We know that the length of the hypotenuse is the square root of the sum of the squares of the other two sides. This means that the length of the hypotenuse is greater than the sum of the other two sides. This is a contradiction, and so we know that the Pythagorean theorem must be true.
Solved Examples
Here are a few examples of using the Pythagorean theorem formula:
Example 1:
A ladder is leaning against a wall and the base of the ladder is 6 meters away from the wall. The ladder is 8 meters long. What is the distance between the top of the ladder and the ground?
Solution:
Let x be the distance between the top of the ladder and the ground.
Using the Pythagorean theorem formula, we have:
x^2 + 6^2 = 8^2
x^2 + 36 = 64
x^2 = 28
x = sqrt(28) ≈ 5.29
Therefore, the distance between the top of the ladder and the ground is approximately 5.29 meters.
Example 2:
A right-angled triangle has one side of length 5 cm and another side of length 12 cm. What is the length of the hypotenuse?
Solution:
Let c be the length of the hypotenuse.
Using the Pythagorean theorem formula, we have:
5^2 + 12^2 = c^2
25 + 144 = c^2
169 = c^2
c = sqrt(169) = 13
Therefore, the length of the hypotenuse is 13 cm.
Example 3:
A square field has a diagonal of length 10 meters. What is the area of the field?
Solution:
Let s be the length of each side of the square field.
Using the Pythagorean theorem formula, we have:
s^2 + s^2 = 10^2
2s^2 = 100
s^2 = 50
s = sqrt(50) ≈ 7.07
Therefore, each side of the square field is approximately 7.07 meters long, and the area of the field is:
Area = s^2 = (sqrt(50))^2 = 50 square meters.
