What is the Pythagorean Theorem formula proof and solved examples
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.
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Pythagorean Theorem Derivation - 1
Let NP be perpendicular to the side MO.
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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 and Explanation
1. What is the Pythagorean Theorem formula?
The Pythagorean Theorem formula is a² + b² = c², where a and b are the legs of a right triangle and c is the hypotenuse.
- a and b are the perpendicular sides.
- c is the longest side opposite the right angle.
- The formula applies only to right-angled triangles.
2. What does the Pythagorean Theorem state?
The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. In symbols: a² + b² = c². This relationship holds true only when one angle is exactly 90°.
3. How do you use the Pythagorean Theorem to find the hypotenuse?
To find the hypotenuse, substitute the known side lengths into a² + b² = c² and solve for c.
- Step 1: Square both legs.
- Step 2: Add the squares.
- Step 3: Take the square root to find c.
4. How do you find a missing leg using the Pythagorean Theorem?
To find a missing leg, rearrange the formula to a² = c² − b² or b² = c² − a².
- Step 1: Square the hypotenuse.
- Step 2: Subtract the square of the known leg.
- Step 3: Take the square root.
5. Can you give an example of the Pythagorean Theorem?
A common example of the Pythagorean Theorem is the 3-4-5 triangle.
- 3² = 9
- 4² = 16
- 9 + 16 = 25
6. What is a Pythagorean triple?
A Pythagorean triple is a set of three positive integers that satisfy a² + b² = c². Examples include:
- 3, 4, 5
- 5, 12, 13
- 8, 15, 17
7. Why does the Pythagorean Theorem only work for right triangles?
The Pythagorean Theorem only works for right triangles because it depends on the presence of a 90° angle. The relationship a² + b² = c² is derived from geometric properties unique to right-angled triangles and does not hold for acute or obtuse triangles.
8. How do you check if a triangle is a right triangle using the Pythagorean Theorem?
To check if a triangle is right-angled, verify whether a² + b² = c² using the longest side as c.
- Step 1: Identify the longest side.
- Step 2: Square all three sides.
- Step 3: Check if the sum of the two smaller squares equals the largest square.
9. What are the real-life applications of the Pythagorean Theorem?
The Pythagorean Theorem is used to calculate distances and lengths in real life. Common applications include:
- Finding the shortest distance between two points (distance formula).
- Construction and architecture measurements.
- Navigation and GPS calculations.
- Computer graphics and engineering design.
10. What are common mistakes when using the Pythagorean Theorem?
A common mistake when using the Pythagorean Theorem is not identifying the hypotenuse correctly. Key points to remember:
- The hypotenuse is always the longest side.
- Use the formula only for right triangles.
- Do not forget to take the square root at the final step.
- Ensure all side lengths use the same units.
