Question

3 from unit 1, lesson 5 this diagram is a straightedge and compass construction of the bisector of angle bac. only angle bac is given. explain the steps of the construction in order. include a step for each new circle, line, and point. 4 from unit 1, lesson 5 this diagram is a straightedge and compass construction of a line perpendicular to line ab passing through point c. which segment has the same length as segment ea? a ec b ed c be d bd

Explanation:

Step1: Angle - bisector construction steps

1. Place the compass at point \(A\). Draw an arc that intersects the two rays of \(\angle BAC\) at points \(B\) and \(C\).
2. Without changing the compass - width, place the compass at \(B\) and draw an arc in the interior of the angle.
3. Then place the compass at \(C\) and draw another arc of the same radius in the interior of the angle. Let these two arcs intersect at point \(D\).
4. Draw the ray \(AD\). Ray \(AD\) is the angle - bisector of \(\angle BAC\).

Step2: Perpendicular - line construction analysis

In the construction of a line perpendicular to \(AB\) through point \(C\), we use the properties of circles. When we construct the perpendicular line, we create circles centered at points on \(AB\) (usually symmetric about \(C\)). In a straight - edge and compass construction of a perpendicular line through a point \(C\) on a line \(AB\), we create two circles of the same radius centered at points on \(AB\) on either side of \(C\). The intersection points of these circles are used to draw the perpendicular line. Since the circles are of the same radius, if we consider a circle centered at \(A\) and a circle centered at \(C\) (in the process of constructing the perpendicular), the segments \(EA\) and \(EC\) are radii of the same - sized circles (or congruent circles in the construction). So \(EA = EC\).

Answer:

1. Steps of angle - bisector construction:
2. With compass at \(A\), draw an arc intersecting the sides of \(\angle BAC\) at \(B\) and \(C\).
3. With compass at \(B\), draw an arc in the interior of the angle.
4. With compass at \(C\), draw an arc in the interior of the angle to intersect the previous arc at \(D\).
5. Draw ray \(AD\) which is the angle - bisector.
6. A. \(EC\)