Compass & Straightedge: Classical Geometric Constructions

**Constructing a Perpendicular Bisector — Guided Entry** Below is segment $\overline{AB}$. Your goal: find its midpoint $M$ and draw the line perpendicular to $\overline{AB}$ through $M$ — using only compass and straightedge. ```geometry {"points":[{"id":"A","x":-4,"y":0},{"id":"B","x":4,"y":0}],"segments":[{"from":"A","to":"B"}],"xRange":[-6,6],"yRange":[-4,4]} ``` **Scaffolded Steps (follow in order):** 1. Open your compass to MORE than half of $AB$. (It must be more than half — why?) 2. Place the compass point on $A$. Draw an arc above AND below $\overline{AB}$. 3. WITHOUT changing the compass width, place the point on $B$. Draw arcs above and below that cross the first arcs. 4. Label the two intersection points $P$ (above) and $Q$ (below). 5. Use your straightedge to draw $\overleftrightarrow{PQ}$. Label the point where $\overleftrightarrow{PQ}$ crosses $\overline{AB}$ as $M$. **Check:** Measure $AM$ and $MB$ with a ruler. Are they equal? Measure the angle at $M$ with a protractor — is it $90°$? **Patty-paper alternative:** Fold the paper so that $A$ lands exactly on $B$. The crease IS the perpendicular bisector. Unfold and compare to your compass construction. **Reflection prompt (choose one):** *Why must the compass width be more than half of $AB$?* OR *How do you know $P$ and $Q$ are both equidistant from $A$ and $B$?*

**Constructing an Angle Bisector — Guided** Below is $\angle BAC$. Construct the ray from $A$ that divides $\angle BAC$ into two equal angles. ```geometry {"figureType":"triangle","angles":{"A":64,"B":90},"vertexLabels":["B","A","C"],"angleLabels":{"A":"64°"},"showAngles":true,"title":"Angle A — Construct the Bisector"} ``` **Scaffolded Steps:** 1. Place compass point at vertex $A$. Draw an arc that crosses BOTH rays $\overrightarrow{AB}$ and $\overrightarrow{AC}$. Label the crossings $D$ (on $\overrightarrow{AB}$) and $E$ (on $\overrightarrow{AC}$). 2. Place compass point on $D$. Draw an arc in the interior of the angle. 3. Keep the SAME compass width. Place point on $E$. Draw an arc crossing the one from Step 2. Label the intersection $F$. 4. Draw ray $\overrightarrow{AF}$. This is the angle bisector. **Check:** Use a protractor to measure $\angle BAF$ and $\angle FAC$. Are they each $32°$? **Patty-paper alternative:** Fold so ray $\overrightarrow{AB}$ lies on top of ray $\overrightarrow{AC}$. The crease from $A$ is the bisector. **Why it works (fill in):** $AD = AE$ because ___. $DF = EF$ because ___. So $\triangle ADF \cong \triangle AEF$ by ___, which means $\angle DAF = \angle EAF$.