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In a right triangle, one side is the base, the longest side is called the hypotenuse and the third side is the height of the triangle.

Pythagorean Theorem:

In the above right triangle, the base is denoted by ‘b’, height by ‘a’ and the hypotenuse by ‘c’.

Applying the Pythagorean theorem, we get

a

Proof:

We need to prove that: AC

Let us draw a perpendicular, BD $\perp$ AC

Now, $\bigtriangleup$ ADB is similar to $\bigtriangleup$ ABC

(According to the theorem, If a perpendicular is drawn from the vertex at the right angle of a right triangle to the hypotenuse then triangles on both sides of the perpendicular are similar to the whole triangle and to each other).

So, $\frac{AD}{AB}$ = $\frac{AB}{AC}$ (sides are proportional)

AD. AC = AB

Also, $\bigtriangleup$ BDC is similar to $\bigtriangleup$ ABC

$\frac{CD}{BC}$ = $\frac{BC}{AC}$

CD. AC = BC

Adding (1) and (2)

AD. AC + CD.AC = AB

AC (AD + CD) = AB

AC.AC = AB

AC

Hence the Pythagorean Theorem is proved.

To find the hypotenuse of a right triangle, we need to find the square root of the sum of the squares of the base and height (the two shorter legs). The hypotenuse is the longest side of a right triangle.

c

c = $\sqrt{a^{2}+b^{2}}$

If one side of a right triangle is 12 and its hypotenuse is 15, what is the length of the other side?

Given hypotenuse c = 15

and side a =12

Applying the Pythagorean theorem,

=>a

=>12

=>144 + b

subtract 144 on both sides

=>b

square root both sides

=>b=9

Length of other side is 9 in

Example 2:

Given two sides of the right triangle are 3 and 4 , what is the length of the hypotenuse?

Solution:

Given side a=3

and side b=4

Applying the Pythagorean theorem,

=>c^{2} =a^{2} + b^{2}

=>c^{2}=3^{2}+4^{2}

=>c^{2} =9+16

=>c^{2}=25

=>c=$\sqrt{25}$

=>c=5

The hypotenuse is 5

=>c=5

The hypotenuse is 5