Unit 3: Bisectors Worksheet Packet - LS Home Page - Free Printable

Let's solve each problem step by step based on the image you provided. The topic is bisectors, which means lines or segments that divide another segment into two equal parts.

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Problem 1: Mark the equal segments with tick marks

We are given three diagrams where a segment bisects another. We need to mark the equal parts using tick marks (small marks on the segments to show they are congruent).

#### A. $\overline{AB}$ bisects $\overline{CD}$

- This means point $A$ lies on segment $\overline{CD}$, and $\overline{AB}$ passes through $C$ and $D$, splitting $\overline{CD}$ into two equal parts.
- Since $\overline{AB}$ bisects $\overline{CD}$, it divides $\overline{CD}$ at its midpoint.
- So, the two parts of $\overline{CD}$ — from $C$ to the intersection point and from the intersection point to $D$ — are equal.
- Mark: Put one tick mark on $CA$ and one on $AD$ (assuming $A$ is the midpoint). But wait — actually, looking at the diagram:
- Points: $C$, $A$, $D$ lie on a line, with $A$ between $C$ and $D$.
- Arrow shows direction from $C$ to $B$, but $B$ is not on $\overline{CD}$.
- So, likely, $\overline{AB}$ intersects $\overline{CD}$ at point $A$, and since it *bisects* $\overline{CD}$, then $A$ must be the midpoint of $\overline{CD}$.
- Therefore, $CA = AD$
- ✔ Mark both segments $CA$ and $AD$ with one tick mark each to indicate they are equal.

#### B. $\overline{EF}$ bisects $\overline{GH}$

- From the diagram: $G$ and $H$ are endpoints of vertical segment; $F$ and $E$ form a horizontal segment crossing at right angles.
- The intersection point is the midpoint of $\overline{GH}$.
- So, $GF = FH$
- ✔ Mark $GF$ and $FH$ with one tick mark each.

#### C. $\overline{LM}$ bisects $\overline{NP}$

- Two intersecting lines: $\overline{NP}$ and $\overline{LM}$ cross at a point.
- Since $\overline{LM}$ bisects $\overline{NP}$, the intersection point is the midpoint of $\overline{NP}$.
- So, $NL = LP$
- ✔ Mark $NL$ and $LP$ with one tick mark each.

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Problem 2: $\overline{WX}$ and $\overline{YZ}$ bisect each other

- Diagram: Horizontal segment $\overline{WX}$, vertical segment $\overline{YZ}$, intersecting at point $Y$ (but labeled as $Y$ above and $Z$ below).
- Wait — if $\overline{YZ}$ is vertical and goes from $Y$ to $Z$, and $\overline{WX}$ is horizontal from $W$ to $X$, they intersect at a point, say $O$.
- But the label says "Y" is above, "Z" below — so $\overline{YZ}$ is vertical.
- They bisect each other, meaning they intersect at their midpoints.
- So, the intersection point is the midpoint of both $\overline{WX}$ and $\overline{YZ}$.
- Therefore:
- $WO = OX$
- $YO = OZ$
- ✔ Mark $WO$ and $OX$ with one tick mark each.
- ✔ Mark $YO$ and $OZ$ with one tick mark each.

> Note: If they bisect each other, the intersection point is the midpoint of both segments.

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Problem 3: $\overline{CD}$ bisects $\overline{AB}$ at $E$, and $AB = 20$

Diagram: Segment $\overline{AB}$ horizontal, $\overline{CD}$ crosses it at point $E$, and $E$ is the midpoint of $\overline{AB}$ because $\overline{CD}$ bisects $\overline{AB}$ at $E$.

So:

#### A. $AE =$ _______

Since $E$ is the midpoint of $\overline{AB}$, and $AB = 20$,

$$
AE = \frac{AB}{2} = \frac{20}{2} = 10
$$

✔ Answer: $AE = 10$

#### B. If $AB = x + 4$, then $x =$ _______

We know $AB = 20$, so:

$$
x + 4 = 20 \\
x = 20 - 4 = 16
$$

✔ Answer: $x = 16$

#### C. If $EB = 4y + 2$, then $y =$ _______

We know $EB = AE = 10$, since $E$ is the midpoint.

So:

$$
4y + 2 = 10 \\
4y = 8 \\
y = 2
$$

✔ Answer: $y = 2$

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✔ Final Answers:

#### Problem 1:
- A. Mark $CA$ and $AD$ with one tick mark each.
- B. Mark $GF$ and $FH$ with one tick mark each.
- C. Mark $NL$ and $LP$ with one tick mark each.

#### Problem 2:
- Mark $WO$ and $OX$ with one tick mark each.
- Mark $YO$ and $OZ$ with one tick mark each.

#### Problem 3:
- A. $AE = 10$
- B. $x = 16$
- C. $y = 2$

Let me know if you'd like a visual sketch explanation!