Pythagoras Theorem
In a
right-angled triangle, the side opposite to the right angle is called the
hypotenuse. It is the longest side of the right-angled triangle. In the following
figure, the side AB, which is opposite to ∠C (right angle), is the hypotenuse.
Pythagoras theorem
relates the lengths of the three sides of a right-angled triangle.
According to
Pythagoras theorem,
“In a
right-angled triangle, the square of the hypotenuse is equal to the sum of the
squares of the other two sides.”
In ∆ABC, if ∠C = 90°, then AB2 = BC2 + AC2
or c2 = a2 + b2
Example 1: In ∆ABC, ∠C = 90°, AC = 8 cm, and BC = 6 cm. Find the length
of AB.
Solution:
∠C = 90° (given)
c2
= a2 + b2 (Pythagoras Theorem)
= (6)2 + (8)2 = 36 + 64 = 100
∴ c = √100 = 10
∴ The length of AB is 10 cm.
Example 2: In ∆PQR, ∠P = 90°, PQ = 24 cm, and QR = 25 cm. Find
the length of PR.
Solution:
∠P = 90° (given)
QR2
= PQ2 + PR2 (Pythagoras Theorem)
252
= 242 + PR2
PR2
= 252 – 242
= 625 – 576 = 49
∴ PR = √49 = 7 cm
The length
of PR is 7 cm.
The Converse of Pythagoras Theorem
“In a
triangle, if the square of the longest side is equal to the sum of the squares
of the other two sides, then the angle opposite the longest side is a right
angle.”
For any
triangle with sides a, b, and c, if c2 = a2 + b2
, then the angle between a and b measures 90° and the triangle is a
right-angled triangle. This is called the converse of Pythagoras theorem.
Example 3: Determine whether the following
triangle is a right-angled triangle.
Solution: QR2 = 302 =
900
PR2
+ PQ2 = 202 + 212 = 841
∴ QR2 ≠ PR2 + PQ2
Hence, ∆PQR is not a right-angled triangle.
Example 4: The sides of a triangle are 10 cm, 24 cm and 26 cm. Determine whether the triangle is a right-angled triangle or not.
Solution: Here, 102 + 242 = 100 + 576 = 676
Again, 262 = 676
∴ 102 + 242 = 262
In the given triangle, (Base)2 + (Perpendicular)2 = (Hypotenuse)2
Hence, the triangle is a right-angled triangle.