Question

The semi perimeter of a triangle exceeds each of its sides by $ 5,{\text{ 3}} $ and $ 2 $ respectively. What is the perimeter of the triangle?
A. $ 12 $
B. $ 10 $
C. $ 15 $
D. $ 20 $

Answer 1

Hint: First of all we will convert the word statements into forms of the mathematical expressions, assuming the semi-perimeter of the triangle is equal to “x”. Simplify the given data using the concept of semi-perimeter and perimeter for the resultant value.

Complete step-by-step answer:
Let us assume that the semi- perimeter of a triangle be $ = x $ .... (A)
Given that the semi perimeter of a triangle exceeds each of its sides by $ 5,{\text{ 3}} $ and $ 2 $ respectively.
Convert the above word statement in the form of mathematical expression.
The first side $ = x - 5 $
The second side $ = x - 3 $ and
The third side $ = x - 2 $
Now, we know that the perimeter of the triangle is the sum of all the three sides of the triangle.
Perimeter $ = x - 5 + x - 3 + x - 2 $
Simplify the right hand side of the equation –
Perimeter $ = 3x - 10 $
Also, we know that the semi-perimeter is half of the perimeter.
Semi-perimeter $ = \dfrac{{3x - 10}}{2} $
By using equation (A), we can write the above equation as –
  $ x = \dfrac{{3x - 10}}{2} $
Do cross-multiplication and simplify –
 $ \Rightarrow 2x = 3x - 10 $
Take the terms with variable “x” on one side and the constant on the other side. Remember when you move any term from one side to another, the sign of the term also changes. Positive term becomes negative and negative term becomes positive.
 $ \Rightarrow 10 = 3x - 2x $
Simplify the above equation –
 $ \Rightarrow 10 = x $
As per our assumption, semi-perimeter is $ x = 10 $
Perimeter is twice the semi-perimeter.
 $ \Rightarrow 2x = 2 \times 10 $
Simplify the above equation –
 $ \Rightarrow 2x = 10 $
So, the correct answer is “10 units”.

Note: Always remember the relation between semi-perimeter and the perimeter. Perimeter is always twice the semi-perimeter. Perimeter is the sum of all the sides of the closed figure. It is measured in centi-meters, meters or kilometres.