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.