Question

1. the unique point that divides a segment into two congruent segments is known as the (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 midpoint 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{3}{4}) (3) (6\frac{1}{4}) (4) (8\frac{1}{3}
5. in the diagram shown, (m) is the midpoint of (overline{lp}) and (n) is the midpoint of (overline{mp}). which of 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 line segment divides it into two congruent segments. So, the unique point that divides a segment into two congruent segments is the midpoint.

Step2: Analyze question 2

If $\overline{EF}$ bisects $\overline{MN}$ at point $W$, by the definition of a bisector, $MW = NW$ (congruent segments), so $\overline{MW}\cong\overline{NW}$.

Step3: Analyze question 3

If $\overline{TU}$ is the perpendicular bisector of $\overline{RS}$, we know that $\overline{RV}\cong\overline{SV}$, $m\angle TVR = 90^{\circ}$, and $\angle TVR\cong\angle UVS$ (vertical angles). But we cannot conclude that $\overline{TV}\cong\overline{UV}$.

Step4: Solve for $x$ in question 4

If $F$ is the mid - point of $\overline{EG}$, then $EF=\frac{1}{2}EG$. Given $EG = 22$, so $EF = 11$. Set $4x−3=11$. Add 3 to both sides: $4x=11 + 3=14$. Divide both sides by 4: $x=\frac{14}{4}=3\frac{1}{2}$.

Step5: Analyze ratio in question 5

If $M$ is the mid - point of $\overline{LP}$ and $N$ is the mid - point of $\overline{MP}$, let $LN=x$, $MN = y$, and $NP = z$. Since $M$ is the mid - point of $\overline{LP}$, $LM=MP$. And since $N$ is the mid - point of $\overline{MP}$, $MN=NP$. So $LP=LM + MP=2MP$ and $MP=2NP$. Then $LP = 4NP$. The ratio of $NP$ to $LP$ is $1$ to $4$.

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