12 multiple-choice questions, progressively harder.
In a circle of radius 151515 cm, equal chords have perpendicular center distances (2x+1)(2x+1)(2x+1) cm and (5x−11)(5x-11)(5x−11) cm. Find their common full chord length.
Solution
Correct answer: D
Equal chords in the same circle have equal perpendicular distances.
Each distance is 9 cm. Use the radius and that distance to find the half chord.
The full chord is 24 cm.
Points A,B,CA,B,CA,B,C occur in that order around a circle. The consecutive arcs have measures (2x+10)∘(2x+10)^\circ(2x+10)∘, (3x−5)∘(3x-5)^\circ(3x−5)∘, and (4x+13)∘(4x+13)^\circ(4x+13)∘. Find the minor arc measure between AAA and CCC.
Correct answer: B
All three consecutive arcs make a full circle.
The three arcs measure 86, 109, and 165 degrees. The route through B measures 195 degrees, so the other route between A and C is the minor arc. It measures 165 degrees.
Diameter ACACAC bounds a semicircle containing BBB. In that semicircle, the arc measures are (3x+4)∘(3x+4)^\circ(3x+4)∘ for AB^\widehat{AB}AB and (5x+16)∘(5x+16)^\circ(5x+16)∘ for BC^\widehat{BC}BC. Find the major arc ACB^\widehat{ACB}ACB.
Correct answer: C
The given arcs sum to 180 degrees.
The minor arc AB is 64 degrees. The major route from A through C to B is the rest of the circle.
The result is 296 degrees.
In two congruent circles, minor arcs JK^\widehat{JK}JK and LM^\widehat{LM}LM have equal measures. Their chords have lengths (x+9)(x+9)(x+9) cm and (3x−7)(3x-7)(3x−7) cm. Find xxx.
Correct answer: A
Equal minor arcs in congruent circles give equal chord lengths.
Both chord lengths are 17 cm, which are positive.
Two congruent circles have radius 25 units. In the first circle, a chord is 40 units long. In the second circle, a chord is at the same perpendicular center distance. Find the second chord length.
Equal perpendicular center distances in congruent circles give equal chord lengths.
For a check, each half chord is 20 units and the center distance is 15 units. The second circle has that same radius and distance.
Two parallel chords of lengths 48 cm and 40 cm lie on the same side of the center in a circle of radius 29 cm. Find the perpendicular distance between their supporting lines.
The half lengths are 24 cm and 20 cm. Apply the right-triangle relation separately.
Both chords lie on the same side of the center, so subtract their center distances. Since 265 is less than 441, the shorter chord is farther from the center.
The sum would measure a separation for chords on opposite sides. A single center distance is not the distance between the lines.
Two concentric circles have radii 13 cm and 20 cm. A line contains a 10 cm chord of the smaller circle. Find the full chord it contains in the larger circle.
The centers are the same, so the line has the same perpendicular distance from both centers. The smaller half chord is 5 cm.
For the larger circle, compute a new half chord.
The larger full chord is 32 cm.
In a circle of radius 121212 units, a chord has full length 21192\sqrt{119}2119 units. Find its perpendicular center distance.
Half the chord is the square root of 119.
The distance is nonnegative, so use 5 units.
Two nonparallel chords of a circle are drawn, and their perpendicular bisectors intersect at P. What must P be?
The circle center lies on the perpendicular bisector of each chord because it is equally far from that chord's endpoints.
The perpendicular bisectors are nonparallel, so they have just one intersection. That intersection P is the center.
A 12 cm chord lies in a radius-10 circle. Another 12 cm chord lies in a radius-13 circle. Which comparison of their perpendicular center distances is correct?
Both half chords are 6 cm.
The second distance square is larger. Both distances are nonnegative, so the chord in the radius-13 circle is farther from its center.
A similarity carries a circle of radius 202020 cm onto one of radius 303030 cm and carries chord ABABAB, of length 242424 cm, onto chord CDCDCD. Find the perpendicular distance from the second center to CDCDCD.
First find the original distance using the 12 cm half chord.
The similarity factor is three halves and preserves perpendicularity, so it scales the center distance too.
The target distance is 24 cm.
Congruent circles have radius 252525 cm. Equal minor arcs have chord lengths (2x+8)(2x+8)(2x+8) cm and (4x−14)(4x-14)(4x−14) cm. Find their common perpendicular center distance.
Equal minor arcs in congruent circles give equal chords.
Each full chord is 30 cm, so each half is 15 cm.
The distance is 20 cm in each circle.
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