Chapter 9.7, Problem 14PSB

To determine

To show: The altitude to the hypotenuse divides it in the ratio of 1:3 in $30°-60°-90°$ triangle.

Explanation of Solution

Given information :

  $30°-60°-90°$ triangle

Formula used :

In $30°-60°-90°$ triangle, sides are in the ratio $1:\sqrt{3}:2$

Calculation :

In triangle CDB , if DB = x then CD = $x\sqrt{3}$

Now, in triangle CDA ,

CD = $x\sqrt{3}$ then AD = $\sqrt{3}.\left(x\sqrt{3}\right)=3x$

Therefore,

  $\begin{array}{l}\frac{DB}{DA}=\frac{x}{3x}=\frac{1}{3}\\ \Rightarrow DB:DA=1:3\end{array}$

Hence proved