Constructions: Bisecting Angles and Lines

You have a long stick. How could you find the exact midpoint without measuring? List two methods (one with string, one with folding, etc.). Now imagine you have only a ruler and compasses on paper — what would you do?

A construction draws an exact geometric figure using only:

• A straight-edge (ruler, but you don't measure lengths with it — just use the edge).
• A pair of compasses (for drawing circles and arcs).

Today's three constructions:

1. Perpendicular bisector of a line segment (cuts the segment in half AT right angles).
2. Angle bisector (cuts an angle exactly in half).
3. Perpendicular from a point to a line (drops a vertical from a point onto a given line).

Given a segment $AB$. To find a line that cuts $AB$ exactly in half at a right angle:

1. Place the compass point at $A$. Open it to MORE than half the length of $AB$.
2. Draw a large arc that goes above AND below the segment.
3. Without changing the radius, move the compass point to $B$.
4. Draw a second arc that crosses the first arc above AND below.
5. Use the straight-edge to join the two intersection points. This line is the perpendicular bisector.

Given an angle $\angle ABC$ with vertex at $B$. To draw a ray from $B$ that cuts $\angle ABC$ exactly in half:

1. Place the compass point at $B$. Choose any radius.
2. Draw an arc that crosses BOTH sides of the angle. Label the crossings $P$ (on $BA$) and $Q$ (on $BC$).
3. Without changing the radius, place the compass at $P$ and draw an arc inside the angle.
4. Same radius from $Q$; draw another arc inside, crossing the previous one at a point $R$.
5. Draw the ray from $B$ through $R$. This is the angle bisector — $\angle ABR = \angle RBC$.

Given a line $\ell$ and a point $P$ NOT on $\ell$. To drop a perpendicular from $P$ onto $\ell$:

1. Compass on $P$. Open wide enough so an arc crosses $\ell$ at two points; call them $X$ and $Y$.
2. Same (or larger) radius from $X$: draw an arc on the opposite side of $\ell$ from $P$.
3. Same radius from $Y$: draw another arc, crossing the previous one at point $Q$.
4. Draw the line $PQ$. It is perpendicular to $\ell$, and the intersection with $\ell$ is the foot of the perpendicular.

Four quick drills on construction methods.

1. 1 A perpendicular bisector divides a segment into two pieces of length ____.

   Show answer
   Pieces equal each other.Equal halves

2. 2 An angle bisector of a $80^{\circ}$ angle creates two angles of ____.

   Show answer
   $80/2$.$40^{\circ}$ each

3. 3 Bisecting a right angle yields what angle?

   Show answer
   $90/2$.$45^{\circ}$

4. 4 Bisecting an angle TWICE divides it into how many equal parts?

   Show answer
   $2 \times 2$ pieces.$4$ equal angles