Question

geometry test sections 1.1 - 1.5
use proper notation. label the pictures. think about and use the relationships
use the figure at right to answer the questions 1 - 3.

1. what are two other names for $overrightarrow{ab}$?
2. name a pair of opposite rays.
3. what is the intersection of the two planes?

use the labelled number line to answer questions 4 - 6.

1. what point on $overrightarrow{cf}$ is one unit from f?
2. what point is the midpoint of $overline{dh}$?
3. what is the length of $overline{bg}$?

use figure 3 at right to answer question 7. label the diagram for full credit.

1. if $ab = 13$ and $ac = 20$, what is $bc$?

use figure 4 to answer question 8. label the diagram for full credit.

1. if $de = 7$, $ef = 2x + 6$, and $df = 19$ find $x$ and find the length of $ef$.

Explanation:

Step1: Identify ray naming rules

A ray is named by its endpoint first. So two other names for $\overrightarrow{AB}$ are $\overrightarrow{AC}$ and $\overrightarrow{AQ}$ (assuming $C$ and $Q$ are on the same ray - like in the diagram).

Step2: Define opposite rays

Opposite rays share an endpoint and form a straight - line. A pair of opposite rays in the figure could be $\overrightarrow{QR}$ and $\overrightarrow{QO}$.

Step3: Recall plane - intersection property

The intersection of two planes is a line. In the figure, the intersection of the two planes is line $\overleftrightarrow{QR}$.

Step4: Analyze number - line points

On the number line, for $\overrightarrow{CF}$, the point one unit from $F$ is $E$ since $F = 0$ and $E=-1$.

Step5: Calculate mid - point

For the segment $\overline{DH}$, $D=-2$ and $H = 2$. The mid - point formula is $\frac{x_1 + x_2}{2}$. So the mid - point is $\frac{-2 + 2}{2}=0$, which is point $F$.

Step6: Find segment length

For segment $\overline{BG}$, $B=-4$ and $G = 1$. The length is $|1-(-4)|=5$.

Step7: Use segment addition postulate

Given $AB = 13$ and $AC = 20$, by the segment addition postulate $AC=AB + BC$. So $BC=AC - AB=20 - 13 = 7$.

Step8: Apply segment addition postulate

Given $DE = 7$, $EF=2x + 6$, and $DF = 19$. By the segment addition postulate $DF=DE + EF$. So $19=7+(2x + 6)$. Simplify the right - hand side: $19=2x+13$. Subtract 13 from both sides: $2x=19 - 13=6$. Divide by 2: $x = 3$. Then $EF=2x + 6=2\times3+6 = 12$.

Answer:

1. $\overrightarrow{AC},\overrightarrow{AQ}$
2. $\overrightarrow{QR},\overrightarrow{QO}$
3. $\overleftrightarrow{QR}$
4. $E$
5. $F$
6. $5$
7. $7$
8. $x = 3,EF = 12$