Question

For an isosceles right angled triangle having each of equal sides a, find the semi-perimeter.
(it's a question of heron's formula)

Answer 1

given,
equal side = a
let, base=a
s = a+b+c/2
s = a+a+a/2
s= 3a/2

Answer 2

Given,

The triangle is isosceles and right angled.

Length of equal sides = a unit

To find,

The semi perimeter of the triangle.

Solution,

We can simply solve this mathematical problem by using the following mathematical process.

Now, in any right angle triangle, the hypotenuse is the largest side and it will be longer than base/height.

So, the equal sides will be obviously the base and height.

Base = Height = a unit

Hypotenuse = x unit (assume)

By, applying Pythagoras theorem, we get that,

a²+a² = x²

x² = 2a²

x = a√2 unit

Perimeter = a√2+a+a = (2a + a√2)unit

Semi perimeter = (2a+a√2)/2 = √2(a√2 +a)/2 = (a√2+a)/√2 unit

[Now, this value of semi perimeter can be used in Heron's formula, for calculating the area of the given triangle.]

Hence, the semi-perimeter is (a√2+a)/√2 unit