**Pythagorus Theorem:-**

It states ,In a right angled triangle the sum of the squares of the other two sides(not including the hypotenuse) is equal to the square of hypotenuse.

i.e. a^2 + b^2 = c^2

some common Pythagorean triplets are (3,4,5), (5,12,13), (7,24,25), (6,8,10), (8,15,17), (9,40,41), (11,60,61), (12,35,37), (20,21,29), (28,45,53) etc.

Now , what is so special about Pythagorus theorem??

- Its application makes a question”SUPER EASY”.
- There are wide variety of question asked in CAT, XAT, IIFT and other aptitude related exams on this concept and you won’t even realize.

Try out the following questions.

**A ladder leans against a vertical wall. The top of the ladder is 8 m above the ground. When the bottom of the ladder is moved 2 m farther away from the wall, the top of the ladder rests against the foot of the wall. What is the length of the ladder? [CAT 2001]**

Lets make a rough diagram of question

Info we have:-

- X, 8, l will form a Pythagorean triplet.( From (a) ).
- Difference between l and x is of 2 units (X + 2 = l ).

So look for Pythagorean triplet from 8 and difference between the rests of the sides must be 2.

YEAH, you got it, it is (8, 15, 17).

So Length of ladder =17

**The length of a ladder is exactly equal to the height of the wall it is resting against. If lower end of the ladder is kept on a stool of length 3m and the stool is kept 9m away from the wall and upper end of the ladder coincides with the tip of the wall. Then, the height of the wall is _______? [CAT 1995]**

Hope now you will be ready with the idea what we need to do.

Make a rough diagram.

Info:-

- ( L , X - 3, 9 ) form a Pythagorean triplet.
- L = X + 3

I think it will be clear from now on what type of Pythagorean triplet we are looking for.

On in which 9 is a side other than hypotenuse (longest side) and difference between the sides is 3m.

Yes, that is (9, 12, 15). So length of ladder is 15m.

**Two sides of a plot measure 32 m and 24 m and the angle between them is a perfect right angle. The other two sides measure 25 m each and the other three angles are not right angles. What is the area of the plot? [CAT 2001]**

(24, 32, 40) form a triplet.

How we came to know????

Let’s see,

(3,4,5) triplet everyone knows, what if we multiply by 2 in each term.

it will become (6,8,10) which is again a triplet.

So multiply my 8 in (3,4,5) and you will have the triplet (24,32,40).

Now can you guess any triplet which have 25?

Yes, (15, 20, 25) Or REMEMBER “In an isosceles triangle the perpendicular to unequal sides dives the side in to half.

Now, Area of a triangle= 1/2 * b * h

Area of ABD = 1/2 * 32 * 24 = 384 m sq.

Area of CEB =Area of CED= 1/2 * 20 * 15=150m sq.

Total area= 384 + 150 + 150 = 684 m sq.

**Based on the figure below, what is the value of x, if y = 10? [CAT 2001]**

Solution:

∆ADE is a right angled triangle with hypotenuse = 10m.

Triplet with side 10 is (6, 8, 10).

Let’s solve from triplet if solved then okay otherwise will see what happens.

Now x - 3 can either be 6 or 8.

CASE - 1) X-3 = 6 => X = 9

∆ABE is also right angle triangle so,

(X+8)^2 = (x-3)^2 + (x+4)^2

289 ≠ 36+100

so x-3 ≠ 6

Case - 2) X-3 = 8 => X = 11

∆ABE is also right angle triangle so,

(X+6)^2 = (x-3)^2 + (x+4)^2

289 = 64 + 225

So x=11 satisfies the ∆ABE, Final answer is x=11.

**Euclid has a triangle in mind. Its longest side has length 20 and another of its sides has length 10. Its area is 80. What is the exact length of its third side? [CAT 2001]**

Are you going to apply the scary formula, Area= √(s(s-a)(s-b)(s-c)) ?????

Hold on!!! There is a better way.

Ares is 80, can you think of any triangle which has hypotenuse 20m???

yes, It is (20,12,15).

Let’s draw it.

10 Can’t be AC as AC is the hypotenuse of ∆ABC which has to be the longest side. Now we can easily find AC from ∆ABC by applying Pythagoras theorem.

In ∆ABC,

AC^2 = AB^2 + BC^2

AC^2 = 256 + 4 = 260

AC=√260 m

**Question For Practice.**

Q1) Harry Potter bought a triangular piece of land of area 150 m^2. Harry took a piece of rope and measured the two sides of the plot and found the largest side to be 50 m long and another side to be 10 m long. What is the exact length of the third side?

(A) 40√3 m (B) 30√2 m (C) √1560 m (D) 24√2 m

Q2) In the given Quadrilateral ABCD, AB= 40cm, BC=42cm and AD=20cm, ∠ABC=∠ADC=900 , Find the area of quadrilateral.

Q3) The numerical value of the product of the three sides of a right angled triangle having a perimeter of 56cm is 4200. Find the length of the hypotenuse.

Q4) 3- A certain city has a circular wall around it, and this wall has four gates pointing north, south, east and west. A house stands outside the city, 3 km north of the north gate, and it can just be seen from a point 9 km east of the south gate. What is the diameter of the wall that surrounds the city?

a. 6 km b. 9 km c. 12 km d. None of these [CAT 2001]

Q5) The perimeter of a right angle triangle is 40 and the sum of square of its sides is 578, Find the area of the triangle