Question

The length of two adjacent sides of a parallelogram are $17$ cm and $12$cm. one of its diagonals is $25$ cm long. Find the area of the parallelogram. Also find the length of altitude from vertex to the side of length $12$ cm.

Answer 1

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

Area of parallelogram$=2\times$ Area of the triangle formed by the adjacent sides and the diagonal

Sides are $17cm,12cm,25cm$

Semi perimeter(s)$=\frac{12+17+25}{2}=\frac{54}{2}=27$cm

Area $=\sqrt{27(27-12)(27-17)(27-25)}$

$\Rightarrow A=\sqrt{27\cdot 15\cdot 10\cdot 2}$

$\Rightarrow A=\sqrt{3\times 3\times 3\times 3\times 5\times 2\times 5\times 2}$

$\Rightarrow A=3\times 3\times 5\times 2$

$\Rightarrow A=90{cm}^2$

$\therefore$ Area of parallelogram $=2\left(90\right)=180c{m}^2$.

Let the height of the parallelogram be $h$

Given base $b=12cm$

$A=b\times h$

$\Rightarrow 180=12\times h$

$\Rightarrow \frac{180}{12}=h$

$\therefore h=15cm$

$\therefore$ Length of altitude $=h=15cm$.