Final answer:
The questions involve high school level Mathematics, dealing with geometric transformations, the properties of isosceles triangles, tangents and secants to circles, and calculations of lengths involving circles and triangles.
Step-by-step explanation:
The subject matter of these questions is Mathematics, specifically concerning geometric shapes, their properties, and their dimensions. The questions seem appropriate for High School level students. These problems typically involve using geometric relationships and formulas to find lengths and areas.
- When constructing a geometric figure using a homothety transformation with coefficient 0.5 centered at a point O, each point of the original shape is mapped to a new point such that each original point, its image, and the center O are collinear, and the distance from the center to an image point is half the distance from the center to the original point.
- The length of the segment AB in an isosceles triangle with extensions equal to the base length can be found using the properties of isosceles triangles and congruent segments.
- The radius of a circle when given the length of a tangent and a secant from the same external point can be determined by using the Pythagorean theorem because the radius, tangent at the point of tangency, and the segment from the point of tangency to the external point form a right triangle.
- When a segment tangents a circle at one point and extends through the circle's center, the length of any segment from the circle's circumference to an external point can be found by using the properties of the circle and similar triangles.