Question

If for the triangle whose perimeter is $37\mathrm{cm}$ and the length of sides are in G.P. also the length of the smallest side is $9\mathrm{cm}$ then the length of remaining two sides are and

• A. $12,16$
• B. $14,14$
• C. $10,18$
• D. $15,13$

Answer 1

The correct option is A

$12,16$

Explanation for the correct answer:

Step 1: Apply Geometric Progression (G.P.) concept

Let, $\mathrm{a},\mathrm{ar},{\mathrm{ar}}^2$ be the sides of the triangle in G.P.

$\mathrm{a}=9\mathrm{cm}$

Perimeter$=37\mathrm{cm}$

As we know, the perimeter of the triangle is equal to the sum of all the sides of the triangle,

$$\begin{gathered} \Rightarrow \mathrm{a}+\mathrm{ar}+{\mathrm{ar}}^2=37 \\ \Rightarrow 9+9\mathrm{r}+9{\mathrm{r}}^2=37 \\ \Rightarrow 9\mathrm{r}+9{\mathrm{r}}^2+9-37=0 \\ \Rightarrow 9\mathrm{r}+9{\mathrm{r}}^2-28=0 \\ \Rightarrow 9r^2+9r-28=0 \end{gathered}$$

Step 2: Finding the value of $‘r’$

By factorization method of quadratic equation, we have to find $p\mathrm{and}q$ where $\mathrm{p}+\mathrm{q}$ is equal to ‘b’ and $\mathrm{p}\times \mathrm{q}$ is equal to ‘c’.

$$\begin{gathered} \Rightarrow 9{\mathrm{r}}^2+21\mathrm{r}-12\mathrm{r}-28=0 \\ \Rightarrow 3\mathrm{r}(3\mathrm{r}+7)-4(3\mathrm{r}+7)=0 \\ \Rightarrow (3\mathrm{r}-4)=0\mathrm{and}(3\mathrm{r}+7)=0 \\ \Rightarrow 3\mathrm{r}=4\mathrm{and}3\mathrm{r}=-7 \\ \Rightarrow \mathrm{r}=\frac{4}{3}\mathrm{and}\mathrm{r}=\frac{-7}{3} \end{gathered}$$

Since, the sides of a triangle cannot be negative,

$\therefore \mathrm{r}=\frac{4}{3}$

Step 3: Substituting the value of $‘r’$ in $\mathrm{a},\mathrm{ar},{\mathrm{ar}}^2$ to find the sides of the triangle,

$$\begin{gathered} \mathrm{a}=9,\mathrm{ar}=9\left(\frac{4}{3}\right),{\mathrm{ar}}^2=9{\left(\frac{4}{3}\right)}^2 \\ \therefore \mathrm{a}=9,\mathrm{ar}=12,{\mathrm{ar}}^2=16 \end{gathered}$$

Hence, the correct answer is option (A).