Question

The altitude of a triangle is five- thirds the length of its corresponding base. If the altitude is increased by $4$ cm and the base decreased by $2$ cm, the area of the triangle would remain the same. Find the base and altitude of the triangle

Answer 1

Let the length of corresponding base be $x$.

Thus, altitude will be $\frac{5}{3}x$.

Now, it is given that altitude is increased by $4$ cm i.e., $\frac{5}{3}x+4$ and base is decreased by $2$ cm i.e., $x-2$.

The area is same for both.

Therefore, $\frac{1}{2}\times x\times \frac{5}{3}x=\frac{1}{2}\times (\frac{5}{3}x+4)\times (x-2)$

$\Rightarrow \frac{5}{3}{x}^2=\frac{5}{3}{x}^2-\frac{10}{3}x+4x-8$

$\Rightarrow 8=\frac{2}{3}x$

$\Rightarrow x=12$

Thus, length of base is $12$ cm

and length of altitude is $\frac{5}{3}\times 12=20$ cm.