Chapter 10.5, Problem 2WE

(a)

To determine

Use Construction 12 to divide $\overline{AB}$ into four congruent segments.

Explanation of Solution

Given:

  $\overline{AB}$ is 15 cm long.

Calculation:

Draw $\overline{AB}$ 15 cm long.

  

To divide $\overline{AB}$ into 4 congruent segments :

1. Choose any point Z not on $\overline{AB}$ . Draw $\overrightarrow{AZ}$ .
2. Use any radius , start with A as center and mark off R , S, T , and U so that AR=RS=ST=TU.
3. Draw $\overline{UB}$ .
4. At R, S and T construct lines parallel to $\overline{UB}$ , intersecting $\overline{AB}$ in X , Y , and Z.

$\overline{AX},\overline{XY},\overline{YZ},\overline{ZB}$ are congruent parts of $\overline{AB}$ .

  

(b)

To determine

Use Construction 4 to divide $\overline{AB}$ into four congruent segments.

Explanation of Solution

Given:

  $\overline{AB}$ is 15 cm long.

Calculation:

Draw $\overline{AB}$ 15 cm long.

  

To divide $\overline{AB}$ into 4 congruent segments, we first divide it into two equal parts by constructing perpendicular bisector :

1. Using any radius greater than $\frac{1}{2}\overline{AB}$, draw four arcs of equal radii, two with center A and two with center B. Label the points of intersections of these arcs X and Y.
2. Draw $\overleftrightarrow{XY}$ .

$\overleftrightarrow{XY}$ is perpendicular bisector of $\overline{AB}$ .Let it intersects $\overline{AB}$ at M.

  

Now , repeat the above steps , using $\overline{AM}\text{ and }\overline{MB}$ .

  

So, $\overline{AB}$ is divided into four congruent segments since $\overline{AM}=\overline{MB}$ , $\overline{AC}=\overline{CM}\text{ and }\overline{MD}=\overline{DB}$