Question

Which of the following can be the length of the third side of a triangle whose two sides measure $18$ cm and $14$ cm?

• A. $4cm$
• B. $3cm$
• C. $5cm$
• D. $32cm$

Answer 1

The correct option is C

$5cm$

The explanation for the correct option.

Option (c): Determine the third side of a triangle whose two sides measure $18\mathrm{cm}$ and$14\mathrm{cm}$.

The sum of lengths of any two sides of a triangle is always bigger than the $3^{rd}$ side's length.

$\begin{array}{rcl}18+14& >& 3^{rd}\mathrm{side}\\ & \Rightarrow & 32>3^{rd}\mathrm{side}\end{array}$

Hence the possible dimension of $3^{rd}$ side from options would be:

$A=\left\{4,3\mathrm{and}5\right\}...\left(i\right)$

Also, The length difference between any $2$ side of a triangle is always less than the length of the $3^{rd}$ side.

$\begin{array}{rcl}18-14& <& 3^{rd}\mathrm{side}\\ & \Rightarrow & 4<3^{rd}\mathrm{side}\end{array}$

Hence the other possible dimension of $3^{rd}$ side from options would be:

$A\text{'}=\left\{5\mathrm{and}32\right\}...\left(ii\right)$

Therefore, from$\left(i\right)\mathrm{and}\mathit{}\mathit{\left(}ii\mathit{\right)}$ concludes that $A\cap A\text{'}=\left\{5\right\}...\left(iii\right)$ that satisfies both the conditions so,$5\mathrm{cm}$ is the length of the $3^{rd}$side.

Hence, option (c) is correct.

The explanation for the incorrect options:

Option (a), (b), and (d): Since from$\left(iii\right)$, $A\cap A\text{'}=\left\{5\right\}$ and (a), (b), and (d) don't satisfy both $\left(i\right)\mathrm{and}\mathit{}\mathit{\left(}ii\mathit{\right)}$.

Therefore options (a), (b), and (d) do not hold true.

Therefore, options (a), (b), and (d) are incorrect.

Hence, option (c) is correct.