Practice
Arcs and Chords Practice
Fifty original questions on congruent arcs and chords, perpendicular diameters, equal center distances, and chord right triangles.
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Question 1
Explanation
Congruent arcs in the same circle have congruent chords, so the second chord is 7.
- Reasoning
Translate the arc congruence directly into chord congruence.
Question 2
Explanation
Congruent arcs in the same circle have congruent chords, so the second chord is 9.
- Reasoning
Translate the arc congruence directly into chord congruence.
Question 3
Explanation
Congruent arcs in the same circle have congruent chords, so the second chord is 12.
- Reasoning
Translate the arc congruence directly into chord congruence.
Question 4
Explanation
Congruent arcs in the same circle have congruent chords, so the second chord is 15.
- Reasoning
Translate the arc congruence directly into chord congruence.
Question 5
Explanation
Congruent arcs in the same circle have congruent chords, so the second chord is 18.
- Reasoning
Translate the arc congruence directly into chord congruence.
Question 6
Explanation
Congruent arcs in the same circle have congruent chords, so the second chord is 21.
- Reasoning
Translate the arc congruence directly into chord congruence.
Question 7
Explanation
Congruent arcs in the same circle have congruent chords, so the second chord is 24.
- Reasoning
Translate the arc congruence directly into chord congruence.
Question 8
Explanation
Congruent arcs in the same circle have congruent chords, so the second chord is 28.
- Reasoning
Translate the arc congruence directly into chord congruence.
Question 9
Explanation
Congruent arcs in the same circle have congruent chords, so the second chord is 31.
- Reasoning
Translate the arc congruence directly into chord congruence.
Question 10
Explanation
Congruent arcs in the same circle have congruent chords, so the second chord is 36.
- Reasoning
Translate the arc congruence directly into chord congruence.
Question 11
Explanation
A perpendicular diameter bisects the chord.
- Reasoning
Mark two equal half-chords.
Question 12
Explanation
The same theorem bisects both the chord and its intercepted arc.
- Reasoning
Carry the bisection to the curved boundary too.
Question 13
Explanation
Chords equidistant from the center are congruent in the same circle.
- Reasoning
Distance must be perpendicular center-to-chord distance.
Question 14
Explanation
The congruent-arcs/chords theorem works in both directions.
- Reasoning
Use the converse in the same circle.
Question 15
Explanation
A perpendicular center segment bisects the chord, so one right triangle contains c/2.
- Reasoning
Split first, solve, then double if needed.
Question 16
Explanation
Half the chord is √(5²-3²)=4. Double it to obtain 8.
- Reasoning
Use d²+(c/2)²=r², then double the half-chord.
Question 17
Explanation
Half the chord is √(10²-6²)=8. Double it to obtain 16.
- Reasoning
Use d²+(c/2)²=r², then double the half-chord.
Question 18
Explanation
Half the chord is √(13²-5²)=12. Double it to obtain 24.
- Reasoning
Use d²+(c/2)²=r², then double the half-chord.
Question 19
Explanation
Half the chord is √(13²-12²)=5. Double it to obtain 10.
- Reasoning
Use d²+(c/2)²=r², then double the half-chord.
Question 20
Explanation
Half the chord is √(17²-8²)=15. Double it to obtain 30.
- Reasoning
Use d²+(c/2)²=r², then double the half-chord.
Question 21
Explanation
Half the chord is √(17²-15²)=8. Double it to obtain 16.
- Reasoning
Use d²+(c/2)²=r², then double the half-chord.
Question 22
Explanation
Half the chord is √(25²-7²)=24. Double it to obtain 48.
- Reasoning
Use d²+(c/2)²=r², then double the half-chord.
Question 23
Explanation
Half the chord is √(25²-15²)=20. Double it to obtain 40.
- Reasoning
Use d²+(c/2)²=r², then double the half-chord.
Question 24
Explanation
Half the chord is √(25²-20²)=15. Double it to obtain 30.
- Reasoning
Use d²+(c/2)²=r², then double the half-chord.
Question 25
Explanation
Half the chord is √(29²-20²)=21. Double it to obtain 42.
- Reasoning
Use d²+(c/2)²=r², then double the half-chord.
Question 26
Explanation
Half the chord is √(34²-16²)=30. Double it to obtain 60.
- Reasoning
Use d²+(c/2)²=r², then double the half-chord.
Question 27
Explanation
Half the chord is √(37²-12²)=35. Double it to obtain 70.
- Reasoning
Use d²+(c/2)²=r², then double the half-chord.
Question 28
Explanation
Half the chord is √(41²-9²)=40. Double it to obtain 80.
- Reasoning
Use d²+(c/2)²=r², then double the half-chord.
Question 29
Explanation
Half the chord is √(50²-30²)=40. Double it to obtain 80.
- Reasoning
Use d²+(c/2)²=r², then double the half-chord.
Question 30
Explanation
Half the chord is √(61²-11²)=60. Double it to obtain 120.
- Reasoning
Use d²+(c/2)²=r², then double the half-chord.
Question 31
Explanation
Half the chord is 4; d=√(5²-4²)=3.
- Reasoning
Halve the chord before the Pythagorean calculation.
Question 32
Explanation
Half the chord is 8; d=√(10²-8²)=6.
- Reasoning
Halve the chord before the Pythagorean calculation.
Question 33
Explanation
Half the chord is 5; d=√(13²-5²)=12.
- Reasoning
Halve the chord before the Pythagorean calculation.
Question 34
Explanation
Half the chord is 12; d=√(13²-12²)=5.
- Reasoning
Halve the chord before the Pythagorean calculation.
Question 35
Explanation
Half the chord is 15; d=√(17²-15²)=8.
- Reasoning
Halve the chord before the Pythagorean calculation.
Question 36
Explanation
Half the chord is 8; d=√(17²-8²)=15.
- Reasoning
Halve the chord before the Pythagorean calculation.
Question 37
Explanation
Half the chord is 24; d=√(25²-24²)=7.
- Reasoning
Halve the chord before the Pythagorean calculation.
Question 38
Explanation
Half the chord is 20; d=√(25²-20²)=15.
- Reasoning
Halve the chord before the Pythagorean calculation.
Question 39
Explanation
Half the chord is 15; d=√(25²-15²)=20.
- Reasoning
Halve the chord before the Pythagorean calculation.
Question 40
Explanation
Half the chord is 21; d=√(29²-21²)=20.
- Reasoning
Halve the chord before the Pythagorean calculation.
Question 41
Explanation
Set 6x-5=3x+19, giving x=8; either expression gives 43.
- Reasoning
Use congruent chords to create an equation, then evaluate the requested length.
Question 42
Explanation
The halves are equal: 4x+1=7x-20 gives x=7. Each half is 29, so the chord is 58.
- Reasoning
Solve equal halves and remember to double.
Question 43
Explanation
Half the chord is 10. Then d=√(26²-10²)=√576=24.
- Reasoning
Use half-chord 10, not full chord 20.
Question 44
Explanation
Equal center distances make the chords congruent. With half-chord 20 and distance 9, r=√(20²+9²)=21.
- Reasoning
Apply congruence, then build the right triangle.
Question 45
Explanation
Half-chord 12 and distance 5 give radius 13, so the diameter is 26.
- Reasoning
Solve for radius, then answer the requested diameter.
Question 46
Explanation
Perpendicular center line bisects AB, so AM=20. Solve 3x+2=20 to get x=6.
- Reasoning
Turn the full chord into an equal half first.
Question 47
Explanation
Congruent chords have congruent arcs: 5x+6=8x-30, so x=12 and each arc is 66°.
- Reasoning
Use the chord theorem converse to equate arc expressions.
Question 48
Explanation
The longest chord is a diameter, length 2r=20, so 22 is impossible.
- Reasoning
Check geometric feasibility before applying a square root.
Question 49
Explanation
The right triangle contains only one half of the bisected chord.
- Reasoning
Identify the actual triangle sides in the diagram.
Question 50
Explanation
Each chord is 20 units from the center because √(25²-15²)=20. Opposite-side distances add to 40.
- Reasoning
Find each perpendicular center distance and use the stated opposite sides.
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Questions to review
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- Question 1Congruent arcs and chordsEasy
- Question 2Congruent arcs and chordsEasy
- Question 3Congruent arcs and chordsEasy
- Question 4Congruent arcs and chordsEasy
- Question 5Congruent arcs and chordsEasy
- Question 6Congruent arcs and chordsEasy
- Question 7Congruent arcs and chordsEasy
- Question 8Congruent arcs and chordsEasy
- Question 9Congruent arcs and chordsEasy
- Question 10Congruent arcs and chordsEasy
- Question 11Perpendicular diameter theoremEasy
- Question 12Perpendicular diameter theoremEasy
- Question 13Equidistant chordsEasy
- Question 14Chord theorem converseEasy
- Question 15Chord right triangleEasy
- Question 16Chord lengthMedium
- Question 17Chord lengthMedium
- Question 18Chord lengthMedium
- Question 19Chord lengthMedium
- Question 20Chord lengthMedium
- Question 21Chord lengthMedium
- Question 22Chord lengthMedium
- Question 23Chord lengthMedium
- Question 24Chord lengthMedium
- Question 25Chord lengthMedium
- Question 26Chord lengthMedium
- Question 27Chord lengthMedium
- Question 28Chord lengthMedium
- Question 29Chord lengthMedium
- Question 30Chord lengthMedium
- Question 31Center-to-chord distanceMedium
- Question 32Center-to-chord distanceMedium
- Question 33Center-to-chord distanceMedium
- Question 34Center-to-chord distanceMedium
- Question 35Center-to-chord distanceMedium
- Question 36Center-to-chord distanceMedium
- Question 37Center-to-chord distanceMedium
- Question 38Center-to-chord distanceMedium
- Question 39Center-to-chord distanceMedium
- Question 40Center-to-chord distanceMedium
- Question 41Chord algebraHard
- Question 42Bisected chord algebraHard
- Question 43Chord distanceHard
- Question 44Equidistant chord synthesisHard
- Question 45Chord synthesisHard
- Question 46Chord bisection algebraHard
- Question 47Arc-chord algebraHard
- Question 48Chord feasibilityHard
- Question 49Error analysisHard
- Question 50Parallel chord synthesisHard