Given:
A figure of a right triangle whose altitude divides the opposite in two segments of 8 and 4 units.
The measure of the altitude is n.
To find:
The value of n.
Solution:
According to the altitude on hypotenuse theorem, the altitude on the hypotenuse of a right triangle is geometric mean of two segments of the hypotenuse.
Let the altitude h divides the hypotenuse in two parts with measure a and b, then
[tex]\dfrac{h}{a}=\dfrac{b}{a}[/tex]
[tex]h^2=ab[/tex]
[tex]h=\sqrt{ab}[/tex] [Because side length cannot be negative]
In the given figure, the altitude is n and it divides the hypotenuse in two segments of 8 units and 4 units.
Using altitude on hypotenuse theorem, we get
[tex]\dfrac{n}{8}=\dfrac{4}{n}[/tex]
[tex]n^2=4\times 8[/tex]
[tex]n^2=32[/tex]
[tex]n=\sqrt{32}[/tex]
Therefore, the measure of altitude is [tex]n=\sqrt{32}[/tex] units.
