Question

3 from unit 1, lesson 4 this diagram is a straightedge and compass construction. a is the center of one circle, and b is the center of the other. a rhombus is a quadrilateral with four congruent sides. explain why quadrilateral acbd is a rhombus.

Explanation:

1. Recall the definition of a circle: all radii of a circle are congruent.

• For circle with center \( A \), \( AC \) and \( AD \) are radii, so \( AC = AD \).
• For circle with center \( B \), \( BC \) and \( BD \) are radii, so \( BC = BD \).

1. Notice that \( AB \) is a common segment, and since the circles are constructed with each other's centers, \( AC = BC = AD = BD \) (because \( AC \) and \( BC \) are radii of their respective circles, and the circles are congruent as they have each other's centers, so their radii are equal).
2. A rhombus is defined as a quadrilateral with four congruent sides. Since \( AC = BC = AD = BD \), quadrilateral \( ACBD \) has four congruent sides, so it is a rhombus.

Answer:

Quadrilateral \( ACBD \) is a rhombus because \( AC \), \( AD \) are radii of the circle centered at \( A \) (so \( AC = AD \)), \( BC \), \( BD \) are radii of the circle centered at \( B \) (so \( BC = BD \)), and the circles are constructed with each other’s centers (so \( AC = BC = AD = BD \)). A rhombus is a quadrilateral with four congruent sides, so \( ACBD \) meets this definition.