Question

4 practice 4 (from unit 1, lesson 4) a, b, and c are the centers of the three circles. which line segment is congruent to hf? a ab b cd c df d cb

Explanation:

Step1: Analyze circle radii

Since A, B, C are circle centers, radii of same/different circles: \( HF \) is a radius (C is center for circle with \( HF \)), \( CD \): C and D are centers, check if \( CD \) length equals radius (C's circle radius: \( HF \) length, and \( CD \) should be equal as A, B, C form congruent circles? Wait, \( HF \) is radius of circle centered at C, \( CD \): let's see, circles centered at A, B, C – so \( AC = BC = AB \) (equilateral triangle?), and \( CD \): D is center, so \( CD \) should be equal to \( HF \) because \( HF \) is radius of C's circle, and \( CD \) is radius of A's or B's? Wait, no: \( HF \) is from C to F (radius), \( CD \): from C to D. Since A, B, C are centers, and D is on A's circle? Wait, the diagram: A, B, C are centers, so \( AC = BC = AB \) (all radii of their respective circles, so circles are congruent). Then \( HF \) is a radius (C to H or C to F), and \( CD \): C to D. Since D is on A's circle, \( AD = AC \) (radius of A's circle), so \( CD \) should be equal to \( HF \) (radius of C's circle, same as A's circle radius).

Step2: Eliminate other options

• \( AB \): length of \( AB \) is radius? No, \( AB \) is distance between centers A and B, which is equal to radius (since circles are congruent, distance between centers is radius? Wait no, if circles are congruent and intersect, distance between centers is equal to radius. Wait, maybe \( HF \) is radius, \( CD \) is also radius (since C and D: D is center, so \( CD \) is radius of C's circle? Wait, no, let's recheck. The key is congruent segments: \( HF \) and \( CD \) are both radii of congruent circles (since A, B, C are centers of congruent circles, and D is related). So \( CD \) is congruent to \( HF \). Other options: \( AB \) is distance between centers, \( DF \) – no, \( CB \) – no. So the correct segment is \( CD \). Wait, but the options: A. AB, B. CD, C. DF, D. CB. So the answer is B? Wait, no, wait the question is which segment is congruent to \( HF \). Let's see: \( HF \) is a radius (C is center, so \( CH = CF = HF \)? Wait, \( H \) and \( F \) are on C's circle? Wait, the diagram: A, B, C are centers. The circle at C has \( H \) and \( F \) on it, so \( CH = CF = \) radius. Then \( CD \): C to D. D is on A's circle, so \( AD = AC = \) radius (since A's circle has radius equal to C's circle, as A, B, C are centers of congruent circles). So \( CD \) is equal to \( AC \), which is equal to \( HF \) (since \( HF \) is radius of C's circle, \( AC \) is radius of A's circle, and circles are congruent). So \( CD \cong HF \). So the correct option is B. CD.

Answer:

B. CD