Chapter 11, Problem 32CUR

To determine

To Find: Value of $AD,AM$ .

Answer to Problem 32CUR

  $\sqrt{5}$ , $\frac{2}{3}\sqrt{5}$

Explanation of Solution

Given:

Right angled $\Delta ABC$ with hypotenuse $\overline{AC}$ .

  $AC=2\sqrt{2}$

Medians $\overline{AD},\overline{BE}$ are medians intersecting at $M$ .

Representing the given triangle as following:

  

It is an isosceles triangle with hypotenuse, therefore

  $AB=BC=\frac{2\sqrt{2}}{\sqrt{2}}=2$

  $BD=DC=1$

Using Pythagorean Theorem in $\Delta ABD$ :

  $\begin{array}{l}A{D}^2=2^2+1^2\\ \Rightarrow AD=\sqrt{5}\end{array}$

Intersection point of medians divides the median in ratio 2:1.

  $AM=\frac{2}{3}AD=\frac{2}{3}\sqrt{5}$