Question

The perimeter of a $△ABC$ is $37cm$ and the ratio between the lengths of its altitudes is $6:5:4$. The lengths of its sides are

• A. $5cm,9cm$, and $18cm.$
• B. $8cm,11cm$, and $13cm.$
• C. $10cm,12cm$, and $15cm.$
• D. $12cm,15cm$, and $19cm.$

Answer 1

The correct option is C $10cm,12cm$, and $15cm.$

Since, perimeter = sum of sides of triangle
Let the sides be x cm, y cm and (37 - x - y) cm.
Also, let the lengths of altitudes be 6a cm, 5a cm and 4a cm
$\because$ Area of a triangle = $\frac{1}{2}\times base\times altitude$
$\therefore \frac{1}{2}\times x\times 6a=\frac{1}{2}\times y\times 5a=\frac{1}{2}(37-x-y)\times 4a$
$\Rightarrow 6x=5y=148-4x-4y$
$\Rightarrow 6x=5y$ and $6x=148-4x-4y$
$\Rightarrow 6x-5y=0$ and $10x+4y=148$
$\Rightarrow x=\frac{5}{6}y$ and $10x+4y=148$
$\Rightarrow 10\left(\frac{5}{6}y\right)+4y=148$
$\Rightarrow 25y+12y=444$
$\Rightarrow 37y=444$
$\Rightarrow y=12$
$\Rightarrow 37y=444$
$\Rightarrow x=\frac{5}{6}\left(12\right)$
$\Rightarrow x=10$
$\Rightarrow \text{Two sides are 10 cm, 12 cm}$
$\therefore \text{third side=37-10-12=15 cm}$