Question

the steps to construct a triangle with side lengths equal to ab using a compass and a straightedge are shown below. prove that the construction results in an equilateral triangle. use the drop - down menus to complete the proof. step 1: set the compass to the length of ab and draw a circle centered at point a. step 2: without changing the compass opening, draw a circle centered at point b. step 3: label one of the points of intersection of the two circles c and use a straightedge to draw ac and bc. click the arrows to choose an answer from each menu. ac = ab because all choose_ are congruent. ab = bc because all choose_ are congruent. choose_ by the transitive property. △abc is an equilateral triangle because choose_ sides are congruent. all 3 at least 2

Explanation:

Step1: Recall circle - radius property

In the first step, a circle is drawn centered at \(A\) with radius equal to the length of \(\overline{AB}\). In the second step, a circle is drawn centered at \(B\) with the same radius (since the compass opening is not changed). In a circle, all radii of the same circle are congruent. For the circle centered at \(A\), \(\overline{AC}\) and \(\overline{AB}\) are radii of the circle centered at \(A\), so \(\overline{AC}\cong\overline{AB}\) because all radii of a circle are congruent.

Step2: Recall circle - radius property again

For the circle centered at \(B\), \(\overline{AB}\) and \(\overline{BC}\) are radii of the circle centered at \(B\). So \(\overline{AB}\cong\overline{BC}\) because all radii of a circle are congruent.

Step3: Apply transitive property

Since \(\overline{AC}\cong\overline{AB}\) and \(\overline{AB}\cong\overline{BC}\), by the transitive property of congruence, \(\overline{AC}\cong\overline{BC}\).

Step4: Define equilateral triangle

A triangle is equilateral if all three of its sides are congruent. Since \(\overline{AC}\cong\overline{AB}\), \(\overline{AB}\cong\overline{BC}\) and \(\overline{AC}\cong\overline{BC}\), \(\triangle ABC\) is an equilateral triangle because all 3 sides are congruent.

Answer:

1. Radii of a circle
2. Radii of a circle
3. \(\overline{AC}\cong\overline{BC}\)
4. all 3