Chapter 13.4, Problem 33WE

(a).

To determine

To find: The reason for each step prove.

Explanation of Solution

Given information: The sides of the triangle are $\overrightarrow{AB}=2\overrightarrow{DB}$ and $\overrightarrow{BC}=2\overrightarrow{BE}$ .

Diagram:

  

The law of vector addition states that if two vectors are acting simultaneously at a point, then it can be written both in magnitude and direction by their adjacent sides from a point and their sum is equal to the resultant vector represented in both direction and magnitude.

Consider the above figure:

(1).The vectors $\overrightarrow{AB}$ and $\overrightarrow{BC}$ represents the sides $AB$ and $BC$ respectively. According to the above law, the side $AC$ represents the resultant vector $\overrightarrow{AC}$ is:

  $\overrightarrow{AC}=\overrightarrow{AB}+\overrightarrow{BC}$

(2). Substitute $2\overrightarrow{DB}$ for $\overrightarrow{AB}$ and $2\overrightarrow{BE}$ for $\overrightarrow{BC}$ in the above expression.

  $\overrightarrow{AC}=2\overrightarrow{DB}+2\overrightarrow{BE}$

(3). Take out $2$ as common in the right hand side of the above expression.

  $\overrightarrow{AC}=2\left(\overrightarrow{DB}+\overrightarrow{BE}\right)$

(4). The vectors $\overrightarrow{DB}$ and $\overrightarrow{BE}$ represents the sides $DB$ and $BE$ respectively. According to the above law, the side $DE$ represents the resultant vector $\overrightarrow{DE}$ is:

  $\overrightarrow{DE}=\overrightarrow{DB}+\overrightarrow{BE}$

Substitute $\overrightarrow{DE}$ for $\overrightarrow{DB}+\overrightarrow{BE}$ in the expression of point (3).

  $\overrightarrow{AC}=2\overrightarrow{DE}$

(b).

To determine

To find: Prove of part (a) by mid-point theorem.

Explanation of Solution

Given information: The sides of the triangle are $\overrightarrow{AB}=2\overrightarrow{DB}$ and $\overrightarrow{BC}=2\overrightarrow{BE}$ .

Diagram:

  

Midpoint theorem states that if a line joining midpoints of any two sides of a triangle, then it must be parallel to the third side and is half of it.

Consider the above triangle $ABC$ shown in the figure:

Let $D$ and $E$ be the midpoints of sides $AB$ and $BC$ respectively. Then according to mid point theorem the line $DE$ is parallel to the side $AC$ .

  $\begin{array}{c}DE\left|\right|AC\\ DE=\frac{1}{2}AC\\ AC=2DE\end{array}$

Hence, the above theorem is proved.