Converse of Pythagoras Theorem
If in a triangle, the square on the greatest side is equal to the sum of squares on the other two sides, then the triangle is right angled triangle.
In the triangle ABC,
AB2 + AC2 = 112 + 602 = 3721 = 612 = BC2
Hence, ∆ABC is a right angled triangle.
(i) There are special sets of numbers a, b and c that makes the Pythagorean relationship true and these sets of special numbers are called Pythagorean triplets. Example: (3, 4, 5) is a Pythagorean triplet.
(ii) Let k be any positive integer greater than 1 and (a, b, c) be a Pythagorean triplet, then (ka, kb, kc) is also a Pythagorean triplet.
So, it is clear that we can have infinitely many Pythagorean triplets just by multiplying any Pythagorean triplet by k.
We shall now see a few examples on the use of Pythagoras theorem in problems.
In the figure, AB ⊥ AC
a) What type of ∅ is ABC?
b) What are AB and AC of the ∆ABC ?
c) What is CB called as ?
d) If AC = AB then, what is the measure of ∠ B and ∠C ?
a) ∆ABC is right angled as AB ⊥ AC at A.
b) AB and AC are legs of ∆ ABC .
c) CB is called as the hypotenuse.
d) ∠ B + ∠ C = 90º and equal angles are opposite to equal sides. Hence, ∠ B = ∠ C =90º /2 =45º
Can a right triangle have sides that measure 5cm, 12cm and 13cm?
Take a = 5 , b = 12 and c = 13
Now, a 2 + b2 = 52 + 122 = 25 + 144 = 169 = 132 = c2
By the converse of Pythagoras theorem, the triangle with given measures is a right angled triangle.
A 20-feet ladder leans against a wall at height of 16 feet from the ground. How far is the base of the ladder from the wall?
The ladder, wall and the ground form a right triangle with the ladder as the hypotenuse. From the figure, by Pythagoras theorem,
202 = 162 + x2
⇒ 400 = 256 + x2
⇒ x2 = 400 − 256 = 144 = 122
⇒ x =12 feet
Therefore, the base (foot) of the ladder is 12 feet away from the wall.
1. We can construct sets of Pythagorean triplets as follows.
Let m and n be any two positive integers (m > n):
(a, b, c) is a Pythagorean triplet if a = m2 − n2 , b = 2mn and c = m2 + n2 (Think, why?)
Complete the table.
2. Find all
integer-sided right angled triangles with hypotenuse 85.
(x + y)2 − 2xy = 852
Pythagorean triplets with hypotenuse 85.
13, 84, 85
36, 77, 85
40, 75, 85
51, 68, 85
Find LM, MN, LN and also the area of ∆ LON.
From ∆LMO, by Pythagoras theorem,
LM2 = OL2 − OM2
⇒ LM2 =132 −122 =169 −144 = 25 = 52
∴ LM = 5 units
From ∆NMO, by Pythagoras theorem,
MN2 = ON2 − OM2
= 152 −122 = 225 −144 = 81= 92
∴ MN = 9 units
Hence, LN = LM + MN = 5 + 9 = 14 units
Area of ∆LON = 1/2 × base × height
= 1/2 × LN × OM
= 1/2 × 14 × 12 = 84 square units.
∆ABC is equilateral and CD of the right triangle BCD is 8 cm.
Find the side of the equilateral ∆ABC and also BD.
As ∆ABC is equilateral from the figure, AB=BC=AC= (x−2) cm.
From ∆BCD, by Pythagoras theorem
BD2 = BC2 + CD2
⇒ ( x + 2)2 = ( x − 2)2 + 82
x2 + 4x + 4 = x2 − 4x + 4 + 82
⇒ 8 x = 82
⇒ x = 8cm
The side of the equilateral ∆ ABC = 6 cm and BD =10 cm.