Question

A triangle has a perimeter of 9 cm. How many combinations of its sides exist if their length in cm are integers?
A. 1
B. 2
C. 3
D. 4

Answer 1

Hint: To solve this question first we will recall the definition of a triangle, properties of a triangle and the formula for perimeter of a triangle. Then according to the properties of a triangle related to the sides of a triangle that the sides are equal in length, or the sum of two sides must be greater than the third side, we will find possible combinations.

Complete step by step solution:
We have been given that a triangle has a perimeter of 9 cm.
We have to find the possible combinations of its sides.
Now, we know that a triangle is a polygon which has three sides and three vertices. We have three different types of triangles based on the lengths of the sides, i.e. equilateral triangle, scalene triangle and isosceles triangle.
Now, we know that the perimeter of a triangle is the sum of the lengths of all three sides.
In an equilateral triangle the measure of all three sides of a triangle must be equal. So if the perimeter of the triangle is 9 cm then there is only one possible combination of length of side i. e. $\left( 3,3,3 \right)$.
Now, we know that in an isosceles triangle two sides of a triangle must be equal in length. So there are possible combinations will be
$\Rightarrow \left( 2,2,5 \right),\left( 4,4,1 \right)$
Now, if the triangle is scalene then no sides are equal in length. So there is any combination possible.
Also we have a property of triangle that the sum of two sides must be greater than third side then the possible combinations will be
$\Rightarrow \left( 2,3,4 \right),\left( 4,4,1 \right)$
So, when we combine all possible combinations we find that there are 4 total possible combinations.
So, the correct answer is “Option A”.

Note: As there is no specific condition given in the question that the length of sides of a triangle are equal in length or not. So we need to find all possible combinations considering the different types of triangles based on the length of sides.