How to find the height of a right triangle

Help Questions

PSAT Math › How to find the height of a right triangle

Questions 1 - 1

Triangle

Note: Figure NOT drawn to scale.

Refer to the above diagram. Give the ratio of the perimeter of $\Delta BDC$ to that of $\Delta ADB$ .

$\sqrt{5}: 1$

$\sqrt{6}:1$

Explanation

The altitude of a right triangle from the vertex of its right triangle to its hypotenuse divides it into two similar triangles.

, as the length of the altitude corresponding to the hypotenuse, is the geometric mean of the lengths of the parts of the hypotenuse it forms; that is, it is the square root of the product of the two:

$BD =\sqrt{ \left ( AD\right ) \left ( DC\right )} = \sqrt{5 \cdot 25} = \sqrt{125} = 5\sqrt{5}$ .

The ratio of the smaller sides of these similar triangles is

The ratio of the smaller side of $\Delta BDC$ to that of $\Delta ADB$ is

$\frac{BD}{AD}=\frac{5\sqrt{5}}{5} = \frac{\sqrt{5}}{1}$ or $\sqrt{5}$ : 1,

so this is also the ratio of the perimeter of $\Delta BDC$ to that of $\Delta ADB$ .