Question

The altitudes of the given triangle are $AP=3,BQ=4$and$CR=5$. What is the ratio of $BC:AC:AB?$

• A. $20:15:12$
• B. $20:12:15$
• C. $15:12:20$
• D. $3:4:5$

Answer 1

The correct option is A $20:15:12$

$\Rightarrow$ $AP=3,BQ=4$ and $CR=5$

Altitudes are in ratio $3:4:5.$

Let area of triangle $=$ L.C.M of $3,4$ and $5=60$

Area of $△ABC$ with base $BC$ and height $AP$.

$\Rightarrow$ Area of $△ABC=\frac{1}{2}\times BC\times AP$

$\Rightarrow$ $60=\frac{1}{2}\times BC\times 3$

$\Rightarrow$ $BC=40$

Area of $△ABC$ with base $AC$ and height $BQ$.

$\Rightarrow$ Area of $△ABC=\frac{1}{2}\times AC\times BQ$

$\Rightarrow$ $60=\frac{1}{2}\times AC\times 4$

$\Rightarrow$ $AC=30$

Area of $△ABC$ with base $AB$ and height $CR$.

$\Rightarrow$ Area of $△ABC=\frac{1}{2}\times AB\times CR$

$\Rightarrow$ $60=\frac{1}{2}\times AB\times 5$

$\Rightarrow$ $AB=24$

$\Rightarrow$ The required ratio $=BC:AC:AB=40:30:24=20:15:12$