Question

1 which of these statements is true? a all rectangles are regular polygons. b all squares are regular polygons. c all rhombuses are regular polygons. d all parallelograms are regular polygons. 2 this diagram is a straightedge and compass construction of a square bacd (not all markings are shown). the construction followed these steps: 1. start with two marked points a and b. 2. use a straightedge to construct line ab. 3. use a previous construction to construct a line perpendicular to line ab passing through a. 4. use a previous construction to construct a line perpendicular to line ab passing through b. 5. use a compass to construct a circle centered at a passing through b. 6. label an intersection point of that circle and the line from the third step as c. 7. use a previous construction to construct a line parallel to line ab passing through c. 8. label the intersection of that line and the line from the fourth step as d. 9. use a straightedge to construct the segments ac, cd, and db. explain why you need to construct a circle in the fifth step.

Explanation:

Step1: Recall the definition of regular polygons

A regular polygon has all - sides equal and all - angles equal.

Step2: Analyze option A

Rectangles have all angles equal to 90°, but their adjacent sides are not always equal. So, not all rectangles are regular polygons.

Step3: Analyze option B

Squares have all four sides equal and all four angles equal to 90°. So, all squares are regular polygons.

Step4: Analyze option C

Rhombuses have all four sides equal, but their angles are not always equal. So, not all rhombuses are regular polygons.

Step5: Analyze option D

Parallelograms have opposite sides equal and opposite angles equal, but not all sides are equal and not all angles are equal. So, not all parallelograms are regular polygons.

For the second - part:
The circle centered at \(A\) passing through \(B\) is constructed to ensure that \(AC = AB\). In a square, all sides are equal. By constructing the circle with radius \(AB\) centered at \(A\), we can find a point \(C\) on the perpendicular line through \(A\) such that the distance from \(A\) to \(C\) is equal to the distance from \(A\) to \(B\). This helps in creating a square where \(AB = AC=CD = DB\).

Answer:

1. B. All squares are regular polygons.
2. The circle is constructed to ensure that one of the sides of the square (\(AC\)) has the same length as the given side (\(AB\)) since all sides of a square are equal.