Question

The sides of a $△ABC$ are $9$ cm, $13$ cm and $12$ cm Find the length of the longest altitude

Answer 1

Side of $\Delta$ABC are:-

$a=9$cm, $b=13$cm, $c=12$cm

$s=\frac{1}{2}(a+b+c)=\frac{1}{2}(9+13+12)=\frac{34}{2}=17$

Area of $\Delta$ABC$=\sqrt{s(s-a)(s-b)(s-c)}$

$=\sqrt{17(17-9)(17-3)(17-12)}$

$=\sqrt{17\times 8\times 4\times 5}$

$=4\sqrt{170}c{m}^2$

If base $=9$cm

Then, area of $\Delta$ABC$=\frac{1}{2}\times$ base $\times$ height

$\Rightarrow$ height $=\frac{2areaof\Delta ABC}{base}$

$=\frac{2.4\sqrt{170}}{9}cm$

$=\frac{8\sqrt{170}}{9}cm$

If base $=13$cm

Height $=\frac{8\sqrt{170}}{13}$cm.

If base $=12$cm

Height $=\frac{8\sqrt{170}}{12}$cm

Length of the highest altitude $=\frac{8\sqrt{170}}{9}$cm.