Question

chapter 1: tools of geometry
directions: answer the following questions to the best of your ability. you are encouraged to write on this test! if you need additional space, staple your work to this sheet. carefully bubble your answers on the scantron.

1. which three points are collinear in the figure to the right?

a a, b, d
b a, b, c
c e, c, a
d f, e, g

1. find the length of $overline{pq}$.

a 50.9 cm
b 46.3 cm
c 25.7 cm
d 21.3 cm
for the next 2 problems, use the figure at the right.

1. name the intersection of the white and grey planes.

a point a
b point g
c $overleftrightarrow{ab}$
d $overleftrightarrow{ce}$

1. name the intersection of lines $overleftrightarrow{ce}$ and $overleftrightarrow{ig}$.

a point a
b point g
c $overleftrightarrow{ab}$
d $overleftrightarrow{ce}$

1. given a is between y and z and $ya = 5.5$, $az = 2x$, and $yz = 41.5$, find $az$.

a 9
b 18
c 36
d 72

1. find the distance between $p(8, 2)$ and $q(3, 5)$.

a 9
b $sqrt{18}$
c $sqrt{34}$
d $sqrt{170}$

1. find the coordinates of the midpoint of $overline{lb}$ if $l(8, 5)$ and $b(-6, 2)$.

a $(1, 3.5)$
b $(2, 1.5)$
c $(7, 3.5)$
d $(7, 1.5)$

1. find the distance rs. (notice the line is increasing by even numbers: 2, 4, 6...)

a - 2 units
b 2 units
c 8 units
d 18 units

Explanation:

Step1: Recall collinear points definition

Collinear points lie on the same line. In the figure, points $A$, $B$, $C$ lie on the same line. So for question 1, the answer is B.

Step2: Use segment - length formula

For question 2, if $PR = 38.3$ cm and $QR=12.6$ cm, then $PQ=PR - QR$. So $PQ = 38.3-12.6=25.7$ cm. The answer is C.

Step3: Identify plane - intersection

For question 3, the intersection of two planes is a line. In the figure, the intersection of the white and grey planes is $\overleftrightarrow{AB}$. The answer is C.

Step4: Find line - intersection

For question 4, the intersection of two lines $\overleftrightarrow{CE}$ and $\overleftrightarrow{IG}$ is a point. From the figure, they intersect at point $G$. The answer is B.

Step5: Apply segment - addition postulate

For question 5, since $A$ is between $Y$ and $Z$, $YA + AZ=YZ$. Given $YA = 5.5$, $AZ = 2x$, and $YZ = 41.5$, we have $5.5+2x=41.5$. Subtract 5.5 from both sides: $2x=41.5 - 5.5=36$, then $x = 18$, and $AZ=2x = 36$. The answer is C.

Step6: Use distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. For points $P(8,2)$ and $Q(3,5)$, $d=\sqrt{(3 - 8)^2+(5 - 2)^2}=\sqrt{(-5)^2+3^2}=\sqrt{25 + 9}=\sqrt{34}$. The answer is C.

Step7: Use mid - point formula

The mid - point formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $(\frac{x_1+x_2}{2},\frac{y_1 + y_2}{2})$. For points $L(8,5)$ and $B(-6,2)$, the mid - point is $(\frac{8+( - 6)}{2},\frac{5 + 2}{2})=(1,3.5)$. The answer is A.

Step8: Calculate distance on number line

For question 8, if $R=-10$ and $S = 8$, then $RS=\vert8-( - 10)\vert=\vert8 + 10\vert = 18$ units. The answer is D.

Answer:

1. B. $A, B, C$
2. C. $25.7$ cm
3. C. $\overleftrightarrow{AB}$
4. B. point $G$
5. C. $36$
6. C. $\sqrt{34}$
7. A. $(1,3.5)$
8. D. $18$ units