Question

If 5,12,135,12,13 are the length of the side of a triangle. show that the triangle is right angled. Find the length of altitude on the hypotenuse.

Answer 1

Let the sides of the triangle be $a=5$, $c=12$ and $b=13$.

$a^2+c^2={\left(5\right)}^2+{\left(12\right)}^2$

$=25+144$

$=169$

$b^2={\left(13\right)}^2$

$=169$

This shows that $a^2+c^2=b^2$.

If the sum of square of two sides of a triangle is equal to the square of the third side, then the triangle is a right angled triangle. (Pythogoras' Theorem)

$\therefore$ the given triangle is a right angled triangle.

To find the altitude on the hypotenuse $\left(h\right)$

We have,

$h^2+x^2=5^2$ $-\left(1\right)$

$h^2+(13-x{)}^2={12}^2$ $-\left(2\right)$

Subtracting $\left(1\right)$ from $\left(2\right)$:

$(13-x{)}^2-x^2={12}^2-5^2$

$169-2\ast 13\ast x+x^2-x^2=144-25$

$169-119=26x$

$x=\frac{50}{26}$

Substituting in $\left(1\right)$

$h^2+{\left(\frac{50}{26}\right)}^2=25$

$h^2=25-\frac{{50}^2}{{26}^2}$

$h^2=\frac{16900-2500}{{26}^2}$

$h^2=\frac{14400}{{26}^2}$

$h^2={\left(\frac{120}{26}\right)}^2$

$h=\left(\frac{120}{26}\right)$

$h$ (height of altitude on the hypotenuse) $=\frac{60}{13}$