Find perimeter, area and altitude of an isosceles triangle given the length of equal sides is $10\mathrm{cm}$ and the base is $14\mathrm{cm}$.

Given:$$a=10\mathrm{cm}$$$$b=14\mathrm{cm}$$

To find: Perimeter, Area and Altitude of an isosceles triangle

Solution: First lets find the perimeter of the given isosceles triangle. $$\begin{array}{rcll}\because & P& =& 2a+b\\ \therefore & P& =& 2\times 10+14\\ \therefore & P& =& 20+14\\ \therefore & P& =& 34\mathrm{cm}\end{array}$$Therefore, the perimeter of the given isosceles triangle is $34\mathrm{cm}$.

Now, lets find area of the given isosceles triangle. $$\begin{array}{lrcl}\because & A& =& \frac{b}{2}\sqrt{a^2-\frac{b^2}{4}}\\ \therefore & A& =& \frac{14}{2}\times \sqrt{{10}^2-\frac{{14}^2}{4}}\\ \therefore & A& =& 7\times \sqrt{100-\frac{196}{4}}\\ \therefore & A& =& 7\times \sqrt{100-49}\\ \therefore & A& =& 7\times \sqrt{51}\\ \therefore & A& =& 7\sqrt{51}{\mathrm{cm}}^2\end{array}$$Therefore, the area of the given isosceles triangle is $7\sqrt{51}{\mathrm{cm}}^2$.

Now, lets find altitude of the given isosceles triangle. $$\begin{array}{lrcl}\because & h& =& \sqrt{a^2-\frac{b^2}{4}}\\ \therefore & h& =& \sqrt{{10}^2-\frac{{14}^2}{4}}\\ \therefore & h& =& \sqrt{100-\frac{196}{4}}\\ \therefore & h& =& \sqrt{100-49}\\ \therefore & h& =& \sqrt{51}\mathrm{cm}\end{array}$$ Therefore, the altitude of the given isosceles triangle is $\sqrt{51}\mathrm{cm}$.