Pythagorean Theorem
The Pythagorean theorem is related to the study of sides of a right angled triangle. It is also called as pythagoras theorem. The pythagorean theorem states that, In a right triangle, (length of the hypotenuse)2 = {(1st side)2 + (2nd side)2}.
In a right angled triangle, there are three sides: hypotenuse, perpendicular and base. The base and the perpendicular make an angle of 90 degree with eachother. So, according to pythagorean theorem:
(Hypotenuse)2 = (Perpendicular)2 + (Base)2
In the above figure1,
c2 = a2 + b2
Therefore, Hypotenuse (c) = √ (a2 + b2)
From the above figure 2, Δ ABC is a right angled triangle at angle C.
From C put a perpendicular to AB at H.
Now consider the two triangles Δ ABC and Δ ACH, these two triangles are similar to each other because of AA similarity. This is because both the triangle have a right angle and one common angle at A.
So by these similarity,
$\frac{a}{c}$ = $\frac{e}{a}$ and $\frac{b}{c}$ = $\frac{d}{b}$
a2 = c*e and b2 = c*d
Sum the a2 and b2, we get
a2 + b2 = c*e + c*d
a2 + b2 = c(e + d)
a2 + b2 = c2 (since e + d = c)
Hence Proved.
According to Euclid, if the triangle had a right angle (90 degree), the area of the square formed with hypotenuse as the side will be equal to the sum of the area of the squares formed with the other two sides as the side of the squares.
From the above figure 3, the sum of the area covered by the two small squares is equal to the area of the third square. Here, a2 is the area of the square ABDE, b2 is the area of the square BCFG and c2 is the area of the square ACHI.
Therefore, a2 + b2 = c2
Hence Proved.
Below are example problems on Pythagorean theorem
Example Problem 1: In a right triangle, the hypotenuse is 5 cm and the perpendicular is 4 cm. Find the length of the base of the triangle?
Solution: By using Pythagoras theorem,
h2 =p2 + b2
52 = 42 + b2
25 = 16 + b2
9 = b2
b = 3
Base is 3 cm
Example Problem 2: In a right triangle, the base is 6 cm and the perpendicular is 8 cm. Find the length of the hypotenuse of the triangle?
Solution: By using Pythagoras Theorem,
h2 =p2 + b2
h2 = 62 + 82
h2 = 36 + 64
h = 10
Hypotenuse is 10 cm
| More topics in Pythagorean Theorem | |
| Converse of Pythagorean Theorem | Pythagorean Identities |
| Pythagorean Triples | Pythagorean Theorem Problems |
