Question

here is a diagram of a straightedge and compass construction. c is the center of one circle, and b is the center of the other. explain why the length of segment bd is the same as the length of segment ab. type your response in the space below.

Explanation:

Step1: Recall compass - construction property

In a compass - construction, when we use a compass to draw circles, the distance between the center of the circle and a point on the circle is the radius of the circle. If we set the compass width to the length of segment \(AB\) and then draw a circle with center \(B\), all points on this circle are at a distance equal to the length of \(AB\) from \(B\).

Step2: Analyze segment \(BD\)

Point \(D\) lies on the circle with center \(B\) that was constructed using the compass set to the length of \(AB\). By the definition of a circle (the set of all points in a plane that are at a given distance from a given point), the length of segment \(BD\) is equal to the radius of the circle with center \(B\). Since the radius of the circle with center \(B\) is set to the length of \(AB\) using the compass, \(BD = AB\).

Answer:

The length of segment \(BD\) is the same as the length of segment \(AB\) because when constructing the circle with center \(B\), the compass was set to the length of \(AB\), and \(D\) is a point on the circle centered at \(B\), so \(BD\) is equal to the radius of that circle which is equal to \(AB\).