Chapter 10.3, Problem 8ST1

To determine

To complete: the blank in the given statement.

Answer to Problem 8ST1

  $4\sqrt{3}$ , $2\sqrt{3}$

Explanation of Solution

Given information:

The given statement is “The medians of equilateral $\Delta ABC$ intersect at point X . if $\overline{AD}$

is a median and $AB=12$ , then $AX=\underset{\_}{}$ and $XD=\underset{\_}{}$ ”.

Consider the below equilateral triangle with given data,

  

In an equilateral triangle, median and altitude are the same.

So, AD is a line perpendicular to BC and divides BC into equal parts.

So, $BD=DC=\frac{AB}{2}=\frac{12}{2}=6$ .

Apply Pythagorean Theorem for right angle triangle $ADC$

  $\begin{array}{l}A{D}^2=A{C}^2-D{C}^2\\ ={12}^2-6^2\\ =144-36\\ =108\\ \Rightarrow AD=6\sqrt{3}\end{array}$

The centroid divides the median in to $2:1$ ratio.

  $\begin{array}{l}AX:XD=2:1\\ \Rightarrow AX=2XD\\ \Rightarrow AX+XD=3XD\\ AD=3XD\\ 6\sqrt{3}=3XD\\ XD=2\sqrt{3}\\ AX=4\sqrt{3}\end{array}$

Hence, $XD=2\sqrt{3}\&AX=4\sqrt{3}$