Question

she completed the following actions to construct the angle bisector of ∠abc, but made a mistake in one of the steps. select the step in which her mistake was made.
step 1: she placed the compass on point b and drew an arc which intersects ray ba at point d and ray bc at point e.
step 2: next, she placed the compass on point d and drew an arc interior of ∠abc.
step 3: then, she placed the compass on point c and drew an arc interior of ∠abc, which intersects the previous arc.
step 4: she labeled the intersection of the two arcs as point f and drew a ray which represents the angle bisector of ∠abc.

Explanation:

Step1: Analyze the correct process of angle - bisector construction

The correct steps for constructing the angle bisector of \(\angle ABC\) are:

1. Place the compass on point \(B\) and draw an arc that intersects \(BA\) at \(D\) and \(BC\) at \(E\).
2. Place the compass on point \(D\) and draw an arc inside the angle.
3. Without changing the compass width, place the compass on point \(E\) and draw an arc that intersects the previous arc.
4. Draw a ray from \(B\) through the intersection point of the two arcs.

Step2: Check each step of the given construction

• Step 1: Correct. This is the first step of the standard angle - bisector construction.
• Step 2: Correct. After getting points \(D\) and \(E\) on the sides of the angle, drawing an arc from \(D\) (with a proper radius) is a valid intermediate step.
• Step 3: Incorrect. In the standard construction, after drawing an arc from \(D\), the next arc should be drawn from \(E\) (not \(C\)) with the same radius as the arc drawn from \(D\).
• Step 4: Since step 3 is wrong, step 4 (which depends on step 3) will also lead to an incorrect bisector. But the root of the mistake is in step 3.

Answer:

Step 3.