Based on the construction seen in the images, let's evaluate each statement:
1. Triangle AXY is an equilateral triangle.
- 1. Since arc XY is drawn centered at B, both points X and Y are equidistant from point B. This means OB = BX = BY, making the radii of the circle centered at B equal.
- 2. Arc BX is congruent to arc BY because they are both radii of the circle centered at B, implying that these arcs subtend equal angles at the center B and thus are equal in length.
- 3. Similarly, arc AX is congruent to arc AY for the same reason – they are both radii of the circle centered at O.
- 4. The solution then states that XY is twice the length of BY multiplied by the cosine of 30°, which equals the square root of 3 times BY. This step uses trigonometric relationships to determine the length of XY based on the known relationship of a 30°-60°-90° triangle, suggesting that angle OBY is 30° and therefore triangle OBY is a 30°-60°-90° right triangle.
- 5. Since AX = AY (radii of the circle centered at O) and XY = √3 BY, and given that BY = BX (radii of the circle centered at B), then AY = AX = XY. This equality of lengths implies that triangle AXY is equilateral, with all sides equal.
Therefore, the conclusion is that triangle AXY is an equilateral triangle based on the given relationships and congruences in the circles with centers O and B.
2. Arc XY is a semicircle.
- If line segment XY is a diameter of the circle (which it appears to be since it passes through the center O), then arc XY is indeed a semicircle. This statement is likely true.
3. Triangle AXY is a right triangle.
- If XY is the diameter of the circle and point A lies on the circumference, then by Thales' theorem, angle AXY is a right angle. Therefore, triangle AXY is a right triangle. This statement is true.
4. Arc BX is congruent to arc BY.
- If point B is on the circle and line segment XY is the diameter, then arc BX and arc BY are both semicircles and hence congruent. This statement is true.
5. Arc AX is congruent to arc AY.
- If point A is equidistant from points X and Y, which lie on the circle, then arcs AX and AY are congruent. However, without knowing if AX and AY are radii or if A lies on the perpendicular bisector of XY, we cannot confirm this statement as true from the image provided.