Question

The area of the triangle shown below is $54\sqrt{6}$square units; find the lengths of the sides.

Short answer

The value ofx is$1.74$.

Step-by-step solution

Step 1. Given Information

The given image is

Step 2. Find the expression of area

• The semiperimeter of the triangle is $\frac{5x+6x+7x}{2}=9x$.
• So, the area of the triangle is$A=\sqrt{9x\times 5x\times 6x\times 7x}=3x^2\sqrt{210}$.

Step 3. Find x

• Equate the given area and the obtained expression and solve to find the value of x.

$\begin{array}{rcl}3x^2\sqrt{210}& =& 54\sqrt{6}\\ x^2& =& \frac{54\sqrt{6}}{3\sqrt{210}}\\ & =& \frac{18}{\sqrt{35}}\\ x& \approx & 1.744\end{array}$