12 multiple-choice questions, progressively harder.
Which description defines a chord of a circle?
Solution
Correct answer: B
A chord is a finite segment whose two distinct endpoints lie on the circle.
An interior segment need not reach the circle, the boundary is curved, and a whole line is not a segment.
In the diagram, which angle is a central angle?
Correct answer: D
A central angle has its vertex at the center O.
The middle letter identifies the vertex, so angle AOB is central. The other two named angles have vertices A and B.
The minor arc AB^\widehat{AB}AB measures 83∘83^\circ83∘. Point CCC lies on the major arc from AAA to BBB. Find m(ACB^)m(\widehat{ACB})m(ACB).
Correct answer: A
The two selected routes fill the circle.
The major arc measures 277 degrees.
Two distinct circles are concentric. Which description must be true?
Correct answer: C
Concentric means that the center is shared.
If the radii were also equal, the point sets would be the same circle. Distinct concentric circles therefore have different radii.
Circle O has radius 8 cm. Which circle must be congruent to it?
Congruent circles have equal radii. A different center can be matched by translation.
The diameter-14 circle has radius 7 cm. The other offered radii are 12 cm and 16 cm, so only radius 8 cm works.
After aligning their centers, what positive dilation factor maps a radius-6 circle onto a radius-15 circle?
Divide target radius by original radius.
The factor is five halves. Multiplying 6 by that factor gives 15.
Point BBB lies on the minor arc from AAA to CCC. The consecutive arcs AB^\widehat{AB}AB and BC^\widehat{BC}BC measure 31∘31^\circ31∘ and 47∘47^\circ47∘. Find m(AC^)m(\widehat{AC})m(AC).
The arcs are adjacent parts of the selected minor arc.
The result is 78 degrees, which is below 180 degrees.
Diameter ACACAC bounds a semicircle containing BBB. Within that semicircle, m(AB^)=147∘m(\widehat{AB})=147^\circm(AB)=147∘. Find m(BC^)m(\widehat{BC})m(BC).
The two arcs complete one semicircle.
The missing arc is 33 degrees.
In one circle, chords ABABAB and CDCDCD have equal lengths. The minor arc AB^\widehat{AB}AB measures 106∘106^\circ106∘. What is the measure of the minor arc CD^\widehat{CD}CD?
The chords are in the same circle, so equal lengths give equal minor arc measures.
A line through center OOO is perpendicular to chord CDCDCD at MMM. If CMCMCM is 666 cm long, find CDCDCD.
The perpendicular through the center bisects the chord. Each half is 6 cm.
The full chord is 12 cm.
A circle has radius 10 cm. A chord is 6 cm from its center, measured perpendicularly. Find the full chord length.
The radius, perpendicular distance, and half chord form a right triangle.
Double the positive half length to get 16 cm.
What is the perpendicular distance from a circle center to any diameter?
A diameter passes through the center.
The distance from a point on a line to that line is zero. This boundary case does not form a nondegenerate center-to-chord right triangle.
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