The figure shows a construction made using technology.

Question

The construction shows angle CAB. A circle was created from point A that intersects with ray AC, creating point E, and ray AB, creating point D. Another point, F, was placed on ray AC between points E and C. A circle was created at point E that has a radius of AF. A circle was created at point D that has a radius of AF. Circles D and E intersect at point G. A ray has been drawn from point A through point G.

Does the construction demonstrate how to bisect an angle correctly using technology? Explain why or why not by describing how it was constructed.

Answer 1

To determine if the construction correctly demonstrates how to bisect angle CAB using technology, we must analyze the steps involved in the construction as described.

1. Circle from Point A: A circle is drawn with center A, intersecting ray AC at point E and ray AB at point D. This step establishes points on each ray that will help in creating congruent segments later.

2. Point F on Ray AC: Point F is placed on ray AC between points E and C. The position of point F is crucial as its distance from point A will be used to create radii for circles centered at points D and E.

3. Circles from Points D and E: Two circles are drawn — one centered at point D with radius AF and another centered at point E with the same radius AF. This means both circles are congruent, having the same radius based on the distance AF.

4. Intersection Point G: The circles from points D and E intersect at point G, creating a new point that is equidistant from both points D and E since G lies on both circles.

5. Ray from A through G: Finally, a ray is drawn from point A through point G.

Now, in terms of angle bisecting:

The construction defines G as a point that lies on both circles, meaning AG will indeed be equidistant from points E and D. Since E and D are points on rays AC and AB respectively, and G is equidistant from these two points, the ray AG should bisect angle CAB.

Hence, the construction successfully demonstrates how to bisect angle CAB using technology by ensuring that the point G, where both circles intersect, lies on the angle bisector originating from point A. Thus, this method produces an accurate angle bisector, confirming that the construction is a correct demonstration for angle bisection.