Question

If each side of a triangle is doubled, then the percentage increase in its area is $200\%$.

• A. True
• B. False

Answer 1

The correct option is B False

Step 1: Find the area of a triangle

Triangle: A three-sided polygon that has three angles, vertices, and sides.

To find the area of the triangle, we will use Heron's formula.

Heron's formula can be given that: $A=\sqrt{s(s-a)(s-b)(s-c)}$

Where $s=\frac{a+b+c}{2}$ and $a,b,c$ are the sides of the triangle.

Step 2: Find the area of the triangle where each side of the triangle is doubled

Now, the sides will be $2a,2b,2c$.

Now, the semi-perimeter of the triangle will be,

$$\begin{gathered} s\text{'}=\frac{2a+2b+2c}{2} \\ \Rightarrow s\text{'}=2\left(\frac{a+b+c}{2}\right) \\ \Rightarrow s\text{'}=2s \end{gathered}$$

Now, the area will be,

$$\begin{gathered} A\text{'}=\sqrt{s\text{'}(s\text{'}-2a)(s\text{'}-2b)(s\text{'}-2c)} \\ \Rightarrow A\text{'}=\sqrt{2s(2s-2a)(2s-2b)(2s-2c)} \\ \Rightarrow A\text{'}=\sqrt{2\left(2\right)\left(2\right)\left(2\right)s(s-a)(s-b)(s-c)} \\ \Rightarrow A\text{'}=4\sqrt{s(s-a)(s-b)(s-c)} \\ \Rightarrow A\text{'}=4A \end{gathered}$$

Step 3: Find the percentage change in the area

The percentage change in area will be,

$$\begin{gathered} =\frac{A\text{'}-A}{A}\times 100 \\ =\frac{4A-A}{A}\times 100 \\ =300\% \end{gathered}$$

Therefore, the percentage increase in area is $300\%$.

Hence, the given statement is False.