Question

1. the unique point that divides a segment into two congruent segments is known as the segments (1) center (2) midpoint (3) dividing point (4) partition point 2. in the diagram shown, it is given that (overline{ef}) bisects (overline{mn}) at point (w). which of the following can we conclude based on this information? (1) (overline{mn}perpoverline{ef}) (2) (overline{ew}congoverline{fw}) (3) (angle ewmcongangle ewn) (4) (overline{mw}congoverline{nw}) 3. in the diagram below, we are told that (overline{tu}) is the perpendicular bisector of (overline{rs}). which of the following can we not conclude based on this information? (1) (overline{tv}congoverline{uv}) (2) (mangle tvr = 90^{circ}) (3) (overline{rv}congoverline{sv}) (4) (angle tvrcongangle uvs) 4. if (f) is the mid - point of (overline{eg}), (eg = 22), and (ef=4x - 3), then which of the following is the value of (x)? (1) (3\frac{1}{2}) (2) (4\frac{1}{4}) (3) (6\frac{1}{4}) (4) (8\frac{1}{2}) 5. in the diagram shown, (m) is the mid - point of (overline{lp}) and (n) is the mid - point of (overline{mp}). which of the following is the ratio of (np) to (lp)? (1) 3 to 1 (2) 1 to 3 (3) 4 to 1 (4) 1 to 4

Explanation:

Step1: Recall mid - point definition

The mid - point of a segment divides it into two congruent segments. So for question 1, the answer is midpoint.

Step2: Analyze bisection property

If $\overline{EF}$ bisects $\overline{MN}$ at point $W$, then $\overline{MW}\cong\overline{NW}$ by the definition of a bisector. For question 2, this is the correct conclusion.

Step3: Understand perpendicular bisector properties

If $\overline{TU}$ is the perpendicular bisector of $\overline{RS}$, we know $\overline{RV}\cong\overline{SV}$, $m\angle TVR = 90^{\circ}$, and $\angle TVR\cong\angle UVS$. But we cannot conclude $\overline{TV}\cong\overline{UV}$. For question 3, this is the non - conclusion.

Step4: Use mid - point formula

If $F$ is the mid - point of $\overline{EG}$, then $EF=\frac{1}{2}EG$. Given $EG = 22$ and $EF=4x - 3$, we have $4x-3=\frac{22}{2}=11$. Solving for $x$:
$4x=11 + 3=14$, so $x=\frac{14}{4}=3.5$. For question 4, the value of $x$ is $3\frac{1}{2}$.

Step5: Calculate segment ratios

Let $LP = 4a$. Since $M$ is the mid - point of $LP$, $LM=MP = 2a$. Since $N$ is the mid - point of $MP$, $NP=a$. So the ratio of $NP$ to $LP$ is $1$ to $4$. For question 5, this is the correct ratio.

Answer:

1. (2) midpoint
2. (4) $\overline{MW}\cong\overline{NW}$
3. (1) $\overline{TV}\cong\overline{UV}$
4. (1) $3\frac{1}{2}$
5. (4) 1 to 4