Final answer:
Statments A (BH is perpendicular to DF) and C (The measure of angle BHD is 90 degrees) are correct because BH is a tangent to circle A at H and tangents are perpendicular to the radius at the point of tangency. Therefore, option D (Both A and C are correct) is the correct answer.
Step-by-step explanation:
In the context of geometric properties of circles, let's tackle the student's question by analyzing each statement:
- BH is tangent to circle A: This implies that the line BH touches the circle at exactly one point and is perpendicular to the radius at the point of tangency.
- DF is a diameter: Being a diameter of the circle, it passes through the center and terminates at both ends of the circle, establishing the longest distance across the circle.
- BH is perpendicular to DF: If BH is tangent to the circle and DF is the diameter that also includes the radius to the point of tangency, then BH is perpendicular to DF at the point where the radius meets the tangent.
- BH is a secant line to the circle: This is incorrect because a secant line intersects the circle in two points, whereas a tangent, as previously established, intersects at only one point.
- The measure of angle BHD is 90 degrees: Since BH is tangent and intersects the diameter (DF) at endpoint H, which is also where the radius meets the tangent, this creates a right angle at BHD.
Therefore, the correct statements are A (BH is perpendicular to DF) and C (The measure of angle BHD is 90 degrees), making option D (Both A and C are correct) the correct answer.