Question

Two sides of an equilateral triangle are reduced by $5$ centimeters and $4$ centimeters. The third side of the triangle is increased by $4$ centimeters to form a right triangle. If the triangle was originally an equilateral triangle, find the length of its sides before the changes were made.

• A. $26cm$
• B. $25cm$
• C. $1cm$
• D. $29cm$

Answer 1

The correct option is B $25cm$

Let the initial length of the side of the equilateral triangle be $xcm.$

Then, the sides of the right triangle will be $(x-5)cm,(x-4)cm,$ and $(x+4)cm.$

We know that $(x+4)cm$ is the largest side, i.e., the hypotenuse.


Applying the Pythagoras theorem, we get
$(x+4{)}^2=(x-4{)}^2+(x-5{)}^2$
$\Rightarrow x^2+16+8x=x^2+16-8x+x^2+25-10x$
$\Rightarrow x^2-26x+25=0$ (Rearranging the equation)
$\Rightarrow x^2-25x-x+25=0$ (Using the splitting method)

Taking x common from the first two terms and $-1$ common from the last two terms, we get
$\Rightarrow x(x-25)-1(x-25)=0$
Taking $(x-25)$ common from both terms,
$\Rightarrow (x-25)(x-1)=0$
$\Rightarrow x=25$ or $x=1$
But, if we take $x=1cm,$ the other two sides will be negative. So, $x=25cm$ is correct.