Question

If every side of a triangle is doubled, the area of the new triangle is $K$ times the area of the old one. Find the value of $K$.

• A. $\sqrt{2}$
• B. $2$
• C. $3$
• D. $4$

Answer 1

Answer: (D)

$4$

Let, the sides of the original triangle be $a,b$ and $c$

so, the semi-perimeter, $s=\frac{a+b+c}{2}$

Original area $=\sqrt{s(s-a)(s-b)(s-c)}$

Now every side is doubled

So, the new sides are $2a,2b$ and $2c$

new semi-perimeter, $s^{\prime }=\frac{2a+2b+2c}{2}=2\left(\frac{a+b+c}{2}\right)=2s$

New area $=\sqrt{s^{\prime }(s^{\prime }-2a)(s^{\prime }-2b)(s^{\prime }-2c)}$

$=\sqrt{\left(2s\right)(2s-2a)(2s-2b)(2s-2c)}$

$=\sqrt{\mathrm{2.2.2.2}s(s-a)(s-b)(s-c)}$

$=4\sqrt{s(s-a)(s-b)(s-c)}$

$=4\times OriginalArea$

Hence, the value of $K$ is $4$.