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Problem 577

# Construct the perpendicular bisector of a given segment. Bisect a given angle.

Expert verified
To construct the perpendicular bisector of a given segment AB, draw intersecting arcs above and below the segment when the compass is placed on points A and B. Then, join the intersecting points with a straightedge. To bisect a given angle ∠XYZ, draw arcs that intersect angle sides from the vertex, draw intersecting arcs from the new points, and draw a ray from the vertex through the intersection of these arcs. This ray is the angle bisector.
See the step by step solution

## Step 1: Construct the perpendicular bisector of a given segment.

To begin, let's name the given segment as AB. We will follow these steps to construct the perpendicular bisector: 1. Place the compass needle on point A and open it to slightly more than half the length of the segment AB. Draw an arc above and below the segment. 2. Without changing the compass opening, place the needle on point B and draw an arc intersecting the previous arcs above and below the segment. 3. Use a straightedge to join the points where the arcs intersect above and below the segment. This line is the perpendicular bisector of segment AB.

## Step 2: Bisect a given angle.

Let's assume we have an angle ∠XYZ that needs to be bisected. Here are the steps to construct the angle bisector: 1. Place the compass needle on the vertex of the angle (point Y) and draw an arc such that it intersects both rays XY and YZ, creating two new points P and Q, respectively. 2. Now, place the compass needle on point P, and draw an arc in the interior of the angle (choose an opening that will allow the arc to be visible). 3. Without changing the compass opening, place the compass needle on point Q and draw another arc intersecting the previous arc you drew in step 2. 4. Use a straightedge to join point Y (vertex) with the point where the two arcs intersect in the interior of the angle ∠XYZ. This newly drawn ray is the angle bisector of ∠XYZ. Now, you have the knowledge to construct both the perpendicular bisector of a segment and bisect a given angle using a compass and a straightedge.

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