Geometry Bring in Focus: PYTHAGORAS

Pythagorean Theorem:

The Pythagorean theorem, also known as 'Pythagoras' theorem, states that, “the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides”. The theorem can also be written as an equation relating the lengths of the sides a, b and c, often called the "Pythagorean equation”. Although Pythagoras theorem is applicable in case of a right-angled triangle only, yet it has a lot of direct and indirect applications.

The basic formula to calculate hypotenuse is a² + b² = c², where c represents the length of the hypotenuse and a and b the lengths of the triangle's other two sides.

If the length of both a and b are known, then c can be calculated as-

Classification of triangles on Pythagoras theorem:

• If a² + b² < c², then the triangle would be an obtuse angled triangle.
• If a² + b² > c², then the triangle would be an acute angled triangle.

Pythagorean Triplets:

In a set of 3 positive integers (a, b, c) which satisfies Pythagoras theorem, the greatest number would always represent the hypotenuse and any of the two smaller numbers could either be the base or hypotenuse.

The following is a list of common Pythagorean triplets, which are asked in the exams directly or indirectly:

(3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (9, 40, 41), (11, 60, 61), (12, 35, 37), (13, 84, 85), (16, 63, 65), (20, 21, 29), (28, 45, 53)

A primitive Pythagorean triplet is one in which a, b and c are co-prime.

Note: If all the numbers in the triplet is multiplied by any constant number, then the resulting numbers would also make a Pythagorean Triplet. As you know that, 3, 4, 5 is a triplet, so when all the three terms are multiplied with any constant number, that will also make a triplet.

Thus, (6,8,10), (9,12,15), (12,16,20), …(30,40,50), all these will be triplets.

Similarly, (10,24,26), (15,36,39), (20,48,52), (25,60,65) will also be Pythagoras triplets as (5,12,13) is a triplet; and all these are multiples of the same. You can learn the application of the Pythagoras theorem in the following examples.

Example 1:

Find the area of triangle with sides 12cm, 35cm, 37cm.

Solution:

You know that it is a Pythagorastriplet, so the hypotenuse must be 37 i.e. longest side.

The rest 2 sides are perpendicular and base.

Area of triangle= ½ × Base × Height
Area of triangle = ½ × 12 × 35 = 210

Example 2:

Find the area of a right-angled triangle of hypotenuse 91 cm and height 35 cm.

Solution:

The given triangle is right angled, so its sides must form a Pythagoras triplet. You know (5, 12, 13) is a triplet, and by observing the lengths of the given side, there’s a common factor i.e. 7. When it is given that hypotenuse is 91 (i.e. 13 × 7) and height is 35 (i.e. 5 × 7), you can say that the base must be 12 × 7 = 84.

Area of triangle = ½ × Base × Height
⇒½ × 91 × 84 = 3822 cm²

Example 3:

Find the area of rectangle ABCD.

Solution:

As you can see from the equations, this question involves complex calculation. When you know (28, 45, 53) is a triplet, then twice of that i.e. (56, 90, and 106) will also be a triplet. As 90 and 106 are already given to us, the third side will be 56 (i.e. 28 × 2) because ∠ADC=90°, as ABCD is a rectangle. So, AD = 56 cm.

Area of rectangle = (56 x 90) = 5040 cm².

Note: Here that having knowledge of the triplets can really make the calculationsin triangle-based questions very easy. Do remember these triplets, those will prove to be handy in papers.