Pythagoras theorem - Definition

For a right angled triangle

Side opposite to right angle is Hypotenuse

Side adjacent to right angle are Base & height

Note :

Here, we can also take AB as base & BC as height.

It does not change our answer

Let   AB = a

BC = b

AC = c

Pythagoras theorem says that

  Square of hypotenuse = Sum of square of other two sides

    a ² + b ² = c ²

There are a lot of interesting things that we can do with Pythagoras theorem

Like

• Pythagoras triplets
• Proof of Pythagoras theorem – The normal way
  
  

Let’s do some questions

Find x

Here,

  ∆ABC is a right angled triangle, right angled at B

And,

  AB = 3, BC = 4, AC = x

By Pythagoras Theorem

AC ² = AB ² + BC ²

x ² = 3 ² + 4 ²

x ² = 9 + 16

x ² = 25

x ² = 5 ²

Cancelling squares

  x = 5

Therefore, x = 5

Find x

Here,

  ∆ABC is a right angled triangle, right angled at B

And,

  AB = 6 cm, BC = 8 cm, AC = x

By Pythagoras Theorem

AC ² = AB ² + BC ²

x ² = 6 ² + 8 ²

x ² = 36 + 64

x ² = 100

x ² = 10 ²

Cancelling squares

  x = 10 cm

Therefore, x = 10 cm

Find x

Here,

  ∆PQR is a right angled triangle, right angled at P

And,

  PR = 8 cm, PQ = 15 cm, QR = x

By Pythagoras Theorem

QR ² = PR ² + PQ ²

Find x

Here,

  ∆PQR is a right angled triangle, right angled at P

And,

  PQ = 24 cm, QR = x, PR = 7 cm

By Pythagoras Theorem

QR ² = PQ ² + PR ²

          x ² = (24) ² + 7 ²

         x ² = 576 + 49

         x ² = 625

         x ² = (25) ²

Cancelling squares

  x = 25 cm

Therefore, x = 25 cm

625 = 5 × 5 × 5 × 5

       = 25 × 25 

       = (25) ²

Find x

In ∆ABC,

  AB = AC

Hence, it is an isosceles triangle.

Now, AD ⊥ BC

In isosceles triangle,

altitude and median are same

∴ AD is median of BC

i.e. D is mid-point of BC

∴ BD = DC = BC/2

   BD = DC = x/2

In ∆ADC, right angled at D.

By Pythagoras Theorem

(AB) ² = (AD) ² + (BD) ²

(37) ² + (12) ² + (x/2) ²

1369 = 144 + x ² /4

1369 − 144 = x ² /4

1225 = x ² /4

   1225 × 4 = x ²

         4900 = x ²

              x ² = 4900

          x ² =  70 ²

Cancelling squares

x = 70 cm

∴ x = 70 cm

Let's look at one last example