Question

1. given segments ab with the length of 2 inches. using a compass and a straightedge, a student constructed arcs with the same radius (more than 1 inch and less than 2 inches in length) using a and b as centers. the intersections of the circles are labeled with points x and y. which of the following statements are true? select all that apply.

a. segments ab and xy are congruent
b. segments ab and xy are perpendicular to each other.
c. segments ab and xy bisects each other.
d. triangles axb and ayb are isosceles triangle and congruent.
e. quadrilateral axby is a square

Explanation:

• Option A: AB is 2 inches, XY's length depends on the compass radius (1 < r < 2). XY will not equal 2 inches, so they are not congruent.
• Option B: The construction of arcs from A and B creates XY as the perpendicular bisector of AB, so AB ⊥ XY.
• Option C: XY bisects AB, and AB bisects XY, so they bisect each other.
• Option D: AX=BX=AY=BY (equal compass radii), so △AXB and △AYB are isosceles. They share side AB, so by SSS congruence, they are congruent.
• Option E: AX=BX=AY=BY, but AB ≠ AX (AB=2, r<2), so AXBY is a rhombus, not a square.

Answer:

B. Segments AB and XY are perpendicular to each other.
C. Segments AB and XY bisects each other.
D. Triangles AXB and AYB are isosceles triangle and congruent.