Constructions and Rigid Transformations

A perpendicular bisector is a line that divides a segment into two equal parts while intersecting it at a right angle (90°). This definition emphasizes both the equality of the segments created and the specific angle of intersection, distinguishing it from other types of lines that may intersect or relate to the segment in different ways.

A rhombus is defined as a quadrilateral, which means it has four sides. The defining characteristic of a rhombus is that all four of its sides are of equal length, making them congruent. This property distinguishes a rhombus from other quadrilaterals, as it does not necessarily have right angles or parallel sides, although it does have opposite angles that are equal and adjacent angles that are supplementary.

To construct a perpendicular line to line ℓ through point P, measuring the distance AB and creating circles at points A and B allows for the precise identification of points C and D, which will define the perpendicular line. This method ensures accuracy by utilizing the properties of circles to find intersections that are equidistant from P, thus guaranteeing that the line CD will be perpendicular to line ℓ at point P. This geometric approach is reliable and effective for constructing perpendicular lines.

To find points equidistant from two locations A and B, one can draw circles with equal radii centered at both points. The intersections of these circles represent locations that are the same distance from A and B, satisfying the condition of equidistance. This geometric method effectively uses the properties of circles, where all points on a circle are at a fixed distance from the center, ensuring that the intersection points are equidistant from both A and B.

An equilateral triangle is defined by having all three sides of equal length, which also means all three angles are equal, each measuring 60°. This congruence in side lengths ensures that the triangle is symmetric and maintains its properties regardless of orientation. The other options describe different types of triangles that do not meet the criteria of having all sides equal, making them incorrect definitions of an equilateral triangle.

Parallel lines are defined as lines that run alongside each other in the same direction, maintaining a constant distance apart without ever meeting. This characteristic ensures that no matter how far they are extended, they will never intersect. This definition is fundamental in geometry and is used in various applications, such as in architecture and design, where maintaining consistent spacing is crucial.

A rectangle is a specific type of quadrilateral characterized by having four right angles, each measuring 90 degrees. This property ensures that the opposite sides are both congruent and parallel, which is fundamental in defining the shape. The presence of these right angles allows for the rectangle to maintain its symmetry and structure, distinguishing it from other quadrilaterals. Thus, the measure of the angles is integral to the definition of a rectangle.

A regular polygon is defined by its equal sides and equal angles, which means that all sides have the same length and all interior angles are of the same measure. This symmetry ensures that the shape is uniform and balanced, making it a fundamental characteristic of regular polygons. Examples include equilateral triangles, squares, and regular hexagons, all of which exhibit this property, confirming that the statement is true.

An isosceles triangle is characterized by having at least two sides that are of equal length. This property distinguishes it from other types of triangles, such as scalene triangles, which have no equal sides, and equilateral triangles, which have all three sides equal. The defining feature of isosceles triangles is the presence of these two equal sides, which also results in two equal angles opposite to them. Hence, the answer is 2, reflecting the two equal sides in an isosceles triangle.

The intersection points of two circles centered at A and B are equidistant from both centers because the circles have equal radii. This means that any point on the circumference of both circles, including the intersection points, is the same distance from the respective centers. Therefore, by definition, these intersection points must be equidistant from both A and B, as they lie on the periphery of both circles at the same radius distance from each center.