Question

Which of the following will form the sides of a triangle?
\[\begin{align}
  & \left( a \right)23cm,17cm,8cm \\
 & \left( b \right)12cm,10cm,25cm \\
 & \left( c \right)6cm,7cm,16cm \\
 & \left( d \right)8cm,7cm,16cm \\
\end{align}\]

Answer 1

Hint: Problems like these are quite a bit straight forward and are easy to solve once we understand all the core concepts behind the problem. We need to have some fair amount of ideas regarding triangles, properties and solutions of triangles to be able to solve this problem. When we are given three sides, then these three sides may or may not form a triangle. For the sides to from a triangle, let us assume that the three sides are ‘a’, ‘b’ and ‘c’, it must satisfy the relation,
\[\begin{align}
  & a+b>c, \\
 & b+c>a, \\
 & c+a>b \\
\end{align}\]
All the above three conditions should meet simultaneously or else the triangle cannot be formed.

Complete step-by-step solution:
Now we start off with the solution to the given problem by one by one checking all the options whether they form a triangle or not.
In the first option we have the sides of the triangle as, \[23cm,17cm,8cm\] . Checking all the options we get,
\[\begin{align}
  & 23+17>8, \\
 & 17+8>23, \\
 & 8+23>17 \\
\end{align}\]
All the conditions satisfy, hence this set of sides can form the triangle. Checking the next option, we have the sides as, \[12cm,10cm,25cm\] , here we have \[12+10<25\] , hence this set of sides cannot form a triangle. Next we have the sides as, \[6cm,7cm,16cm\] , here also \[6+7<16\] , so this set of sides also cannot make a triangle. The next set of sides are, \[8cm,7cm,16cm\] , here too \[8+7<16\] , which violates the relation, hence among all the options, only one forms a triangle.
Option (a) is therefore the correct answer which has the sides that form a triangle.

Note: We need to have some prior knowledge of triangles and their various properties before solving these types of problems. All the relations regarding the triangles should be well remembered or else we won’t be able to proceed with the question. As soon as we see that an option violates the rule of the triangle, we need not to proceed further to check all of the relations, because we must remember that all the three relations should hold true simultaneously.