Question

the diagram shown is a construction using a compass and a straightedge, with points a and b being the centers of the two circles. explain why quadrilateral acbd is a rhombus.

Explanation:

Step1: Recall circle - radius property

In a circle, all radii are equal. In circle with center \(A\), \(AC = AD\) since they are radii of the same circle. In circle with center \(B\), \(BC=BD\) as they are radii of the same circle.

Step2: Consider the distance between centers

Also, \(AC = BC\) because the distance from \(A\) to \(C\) and \(B\) to \(C\) is set by the compass - construction. Similarly, \(AD = BD\). So \(AC=AD = BC = BD\).

Step3: Apply rhombus definition

A rhombus is a quadrilateral with all four sides equal. Since \(AC = AD=BC = BD\), quadrilateral \(ACBD\) is a rhombus.

Answer:

Quadrilateral \(ACBD\) is a rhombus because all four of its sides (\(AC\), \(AD\), \(BC\), and \(BD\)) are equal, which is the definition of a rhombus. The equal - side lengths are a result of the properties of circle radii and the compass - straightedge construction.