How Does the Pythagorean Theorem Work?
One of the well-recognized formulas in modern mathematics is the Pythagorean Theorem, which renders us with the association between the sides in a right triangle. A right triangle has two legs and a hypotenuse. The two legs meet at an angle of 90° while the hypotenuse is the longest side of the right triangle and is that side which is opposite to the right angle. Simply, a Pythagoras equation describes the relationship between the three sides of a right-angled triangle.
The Pythagorean Theorem explains the link in every right triangle is:
a² + b² = c²
Formula For Pythagoras Theorem
The formula for Pythagoras Theorem is given by:
Perpendicular² + Base² = Hypotenuse²
Or
a² + b² = c²
Where a, b and c represents the sides of the right-angled triangle with hypotenuse as c.
Use of Pythagorean Theorem Formula
The Pythagoras theorem is used to calculate the sides of a right-angled triangle. If we are given the lengths of two sides of a right-angled triangle, we can simply determine the length of the 3rd side. (Note that it only works for right-angled triangles!)
The theorem is frequently used in Trigonometry, where we apply trigonometric ratios such as sine, cos, tan; to find out the length of the sides of the right triangle.
Derivation of Pythagorean Theorem
Take into account a right-angled triangle ΔMNO. From the figure shown below, it is right-angled at N.
(Image will be uploaded soon)
Pythagorean Theorem Derivation - 1
Let NP be perpendicular to the side MO.
(Image will be uploaded soon)
Pythagoras Theorem Derivation - 2
From the above-given figure, consider the ΔMNO and ΔMPN,
In ΔMNO and ΔMPN,
∠MNO = ∠MPN = 90°
∠M = ∠M → common
Using the MM criterion for the similarity of triangles we have,
Δ MNO ~ Δ MPN
Thus, MP/MN = MN/MO
⇒ MN² = MO x MP…(1)
Considering ΔMNO and ΔNPO from the figure below:
Pythagorean Theorem Derivation -3
∠O = ∠O → common
∠OPN = ∠MNO = 90°
Applying the principle of the Angle Angle(AA) criterion for the similarity of triangles, we come to the conclusion that,
ΔNPO ~ ΔMNO
Thus, OP/NO = NO/MO
⇒ NO² = MO x OP …..(2)
From the similarity of triangles, we come to the conclusion that,
∠MPN = ∠OPN = 90°
That said, if a perpendicular is constructed from the right triangle of a right-angled vertex to the hypotenuse, then the triangles so formed on both sides of the perpendicular are identical to each other and as well the whole triangle.
To Prove: MO² = MN² + NO²
By adding up the equation (1) and equation (2), we obtain:
MN² + NO² = (MO x MP) + (MO x OP)
MN² + NO² = MO (MP + OP)….(3)
Since MP + OP = MO, substituting the value in equation (3).
MN² + NO² = MO (MO)
Now, it becomes
MN² + NO² = MO²
Therefore, the Pythagorean theorem is proved.
Solved Examples
Example:
Calculate the hypotenuse of a right-angled triangle whose lengths of two sides are 6 cm and 9 cm.
Solution: Given the criteria are:
Perpendicular = 9 cm
Base = 6 cm
Applying the Pythagoras theorem we have
Hypotenuse² = Perpendicular² + Base²
Now, putting the values we have will get:
Hypotenuse² = 9² + 6²
Hypotenuse² = 81 + 36
Hypotenuse =√117
Hypotenuse = √10.8.
Example:
Solve the right-angled triangle with the two given sides 8, b, 17
Solution:
Begin with: a² + b² = c²
Put in the values we know: 8² + b² = 17² = 353
Calculate squares: 64 + b² = 289
Take 64 from both sides: 64 − 64 + b² = 289 − 64
Calculate: b² = 225
Square root of both sides: b = √225
Calculate: b = 15
Example:
Determine the distance of diagonal across a square of size 2?
Solution:
Begin with: a² + b² = c²
Put in the values we know: 2² + 2² = c²
Calculating the squares: 2 + 2 = c²
2 + 2 = 4: 4 = c²
Now, let’s swap the sides: c² = 4
Square root of both sides: c = √4
This is about: 2.
FAQs on Pythagorean Theorem Formula Explained
1. What is the Pythagorean theorem and what is its fundamental formula?
The Pythagorean theorem is a fundamental principle in geometry that states for any 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 (legs). The formula is expressed as a² + b² = c², where 'c' is the hypotenuse and 'a' and 'b' are the other two sides.
2. How is the Pythagorean theorem formula written using different variables like P, B, H?
While a² + b² = c² is common, the formula is often expressed using variables that represent the sides' names. In this notation, 'P' stands for Perpendicular, 'B' for Base, and 'H' for Hypotenuse. The formula becomes H² = P² + B². It's crucial to remember that the hypotenuse (H or c) is always the longest side and is by itself on one side of the equation.
3. Can the Pythagorean theorem formula be used to find the length of a leg (base or perpendicular)?
Yes, the formula can be rearranged to find the length of a leg if the hypotenuse and the other leg are known. To find a missing leg (let's say 'a'), you would use the formula: a² = c² - b². Similarly, to find leg 'b', you would use b² = c² - a². This is a common application of the theorem in solving geometry problems.
4. What are some real-life examples where the Pythagorean theorem is applied?
The Pythagorean theorem has many practical applications in the real world. Some key examples include:
Architecture and Construction: Builders use it to ensure that corners of buildings are perfectly square (90 degrees) by using the 3-4-5 ratio.
Navigation: A ship or airplane can calculate the shortest distance to a destination by treating its north/south and east/west paths as the legs of a right triangle.
Surveying: Surveyors use it to calculate the steepness or gradient of slopes and hills.
Crime Scene Investigation: Investigators can determine the trajectory of a bullet by using the theorem to model its path.
5. What are Pythagorean Triples, and how does the '3-4-5' triangle relate to the theorem?
A Pythagorean Triple is a set of three positive integers (a, b, c) that perfectly satisfy the Pythagorean formula a² + b² = c². The most famous example is the set (3, 4, 5). It demonstrates the theorem because if you take the two smaller numbers as the legs and the largest as the hypotenuse, the equation holds true: 3² + 4² = 9 + 16 = 25, and 5² is also 25. This proves that a triangle with side lengths of 3, 4, and 5 must be a right-angled triangle.
6. Why does the Pythagorean theorem only work for right-angled triangles?
The theorem is derived directly from the unique properties of a 90-degree angle. The relationship where the sum of the squares of the two shorter sides equals the square of the longest side is exclusive to triangles containing a right angle. For acute triangles (all angles less than 90°) or obtuse triangles (one angle greater than 90°), this equality does not hold. For those triangles, a more general formula called the Law of Cosines is needed, which accounts for the different angles.
7. What is the Converse of the Pythagorean Theorem and why is it important?
The Converse of the Pythagorean Theorem states that if the lengths of the three sides of a triangle (a, b, and c) satisfy the equation a² + b² = c², then the triangle must be a right-angled triangle. The angle opposite the longest side 'c' will be the right angle. Its importance lies in its ability to prove whether a given triangle is a right triangle or not, without needing to measure its angles directly. This is a crucial tool for verification in geometry and construction.
8. How do you apply the Pythagorean theorem formula in a sample problem?
Imagine you have a right-angled triangle with a base of 8 cm and a perpendicular of 15 cm. To find the hypotenuse (c), follow these steps:
1. Write the formula: a² + b² = c²
2. Substitute the known values: 8² + 15² = c²
3. Calculate the squares: 64 + 225 = c²
4. Add the results: 289 = c²
5. Find the square root: c = √289 = 17
The length of the hypotenuse is 17 cm.
