Chapter 1.3, Problem 67PPS

(a)

To determine

To Find: The sketch for the three different line segments.

Answer to Problem 67PPS

The required line segments is shown in Figure 1.

Explanation of Solution

By the use of the straightedge the three different line segments is shown in Figure 1.

  

Figure 1

(b)

To determine

To Find: Themidpoint on each of the line segment AB and label it as C and the find the midpoint of the line segment AC and label it as D.

Answer to Problem 67PPS

The required diagram is shown in Figure 2

Explanation of Solution

From the figure 1 find the line segment AB and label the point C on it then find the midpoint AC and label the point D in this way follow the same process for the other segments as shown in Figure 2.

  

Figure 2

(c)

To determine

To Find: The measure of the AB, AC, and AD for each of the line segment and organize the result in the table.

Answer to Problem 67PPS

The required table is shown in Table 1.

Explanation of Solution

From the Figure 2 the measure of the first line segment is $AB=3.6\text{cm}$

  $AC=1.8\text{cm}$ and $AD=0.9\text{cm}$ .

From the Figure 2 the measure of the second line segment is $AB=5.5\text{cm}$

  $AC=2.7\text{cm}$ and $AD=1.4\text{cm}$ .

From the Figure 2 the measure of the third line segment is $AB=7.4\text{cm}$

  $AC=3.7\text{cm}$ and $AD=1.9\text{cm}$ .

The table for the above results is shown in Table 1.

Table 1

Line segment	AB	AC	AD
1	3.6	1.8	0.9
2	5.5	2.7	1.4
3	7.4	3.7	1.9

(d)

To determine

To Find: The measure for the expression for the AC and AD when $AB=x$ .

Answer to Problem 67PPS

The relation is $AD=\frac{1}{4}x$ and $AC=\frac{x}{2}$ .

Explanation of Solution

From the data shown in Table 1 $2AC=AB$ then,

  $\begin{array}{l}2AC=x\\ AC=\frac{x}{2}\end{array}$

From the data shown in Table 1 $2AD=AC$ then,

  $\begin{array}{l}2AD=AC\\ AD=\frac{1}{2}\left(\frac{x}{2}\right)\\ AD=\frac{1}{4}x\end{array}$

Thus, the relation is $AD=\frac{1}{4}x$ and $AC=\frac{x}{2}$ .

(e)

To determine

To Find: The conjecture about the relationship between the AB and each of the segment when you have to continue and determine the midpoints of the segment and the midpoint found previously.

Answer to Problem 67PPS

The length is multiplied by the factor of half every time a midpoint is made this shows that the length of the nth midpoint should be ${\left(\frac{1}{2}\right)}^{x}x=\frac{1}{2^n}x$ .

Explanation of Solution

The part d shows that the length is multiplied by the factor of half every time a midpoint is made this shows that the length of the nth midpoint should be ${\left(\frac{1}{2}\right)}^{x}x=\frac{1}{2^n}x$ .