College Geometry

Question

Three diagrams are given below. 

Based on the above diagrams, which of the following statements are true? 

Create an answer using the numbers associated with the true statements. For example, if only 1, 2, and 5 are true, then the answer is 125; if only 3 and 5 are true, then the answer is 35, etc. 

1.   AB⎯⎯⎯⎯⎯⎯⎯ is a diameter of ⊙P.

2.   ∠APC is a central angle of ⊙P.

3.   If HK⎯⎯⎯⎯⎯⎯⎯⎯⎯⊥GF⎯⎯⎯⎯⎯⎯⎯⎯ in ⊙H, then HJ⎯⎯⎯⎯⎯⎯⎯⎯≅JF⎯⎯⎯⎯⎯⎯⎯.

4.   HK⎯⎯⎯⎯⎯⎯⎯⎯⎯ is a chord of ⊙H.

5.   BC⏜ is a minor arc of ⊙P.

6.   ∠DBA is a central angle of ⊙P.

7.   The circles in Figure 2 are concentric circles.

a) 13457

b) 123457

c) 125

d) 45

e) 2356

f) 23457

g) None of the above

Answer 1

Let's go one by one with these statements:

1) AB is the diameter of of circle P because P is the midpoint of the circle, A and B are points on the circle, and AB goes through P. True

2) Angle APC is a central angle of circle P because A and C are points on the circle and P is the midpoint. The angle is at the center of the circle. True

3) If HK and GF are perpendicular, then HJ = GF. HK is a radius of the circle whereas GF is a chord that cuts through HK. Even if HK and GF were perpendicular, the chord GF could go through any point of radius HK and thus, HJ = GF wouldn't necessarily be true. False

4) HK is a chord of the circle. HK is NOT a chord because a chord by definition has both of its endpoints on the circle itself. HK is the radius of the circle as H is its midpoint. False

5) BC is a minor arc of P. BC is indeed a minor arc of circle P because the arc from B to C spans less than half of the circle's edge. True

6) Angle DBA is a central angle of circle P. Angle DBA is NOT a central angle of circle P, because B is not the midpoint of the circle and the angle isn't formed from the circle's midpoint. False

7) The circles in Figure 2 are concentric. They are NOT concentric, because they both don't share the same midpoint. False

Thus, the only true statements are 1, 2, and 5. The correct answer is c)