CLASS 9 MATHS · CHAPTER 5 · STUDY NOTES
I’m Up and Down, and Round and Round
Understand i’m up and down, and round and round through short explanations, labelled models and original practice. Read one topic, explain it in your own words, then check.
UNDERSTAND · APPLY · CHECK
One centre, one fixed radius
Name the parts of a circle precisely.Radius, diameter and chord
y (units)
x (units)
- Circle · Every boundary point is the same distance from its centre.
- Diameter · A chord through O.
- Radius · Centre to boundary.
- Another chord · It need not pass through O.
- OO · (0, 0)
- 2Radius endpoint · (0, 5)
An ideal circle of radius 5 units. Equal units on both axes.
Read the plotted values
| x | y |
|---|---|
| 5 | 0 |
| 4.957224306869052 | 0.6526309611002579 |
| 4.8296291314453415 | 1.2940952255126037 |
| 4.619397662556434 | 1.913417161825449 |
| 4.330127018922194 | 2.4999999999999996 |
| 3.9667667014561756 | 3.043807145043603 |
| 3.5355339059327378 | 3.5355339059327373 |
| 3.043807145043603 | 3.9667667014561756 |
| 2.5000000000000004 | 4.330127018922193 |
| 1.9134171618254492 | 4.619397662556434 |
| 1.2940952255126037 | 4.8296291314453415 |
| 0.6526309611002585 | 4.957224306869052 |
| 3.061616997868383e-16 | 5 |
| -0.6526309611002569 | 4.957224306869053 |
| -1.2940952255126041 | 4.8296291314453415 |
| -1.9134171618254485 | 4.619397662556434 |
| -2.499999999999999 | 4.330127018922194 |
| -3.043807145043603 | 3.9667667014561756 |
| -3.5355339059327373 | 3.5355339059327378 |
| -3.9667667014561765 | 3.043807145043602 |
| -4.330127018922194 | 2.4999999999999996 |
| -4.619397662556434 | 1.9134171618254494 |
| -4.829629131445341 | 1.294095225512605 |
| -4.957224306869052 | 0.6526309611002579 |
| -5 | 6.123233995736766e-16 |
| -4.957224306869053 | -0.6526309611002566 |
| -4.829629131445342 | -1.2940952255126017 |
| -4.619397662556435 | -1.913417161825446 |
| -4.330127018922193 | -2.5000000000000004 |
| -3.9667667014561756 | -3.043807145043603 |
| -3.5355339059327386 | -3.5355339059327373 |
| -3.0438071450436044 | -3.9667667014561747 |
| -2.500000000000002 | -4.330127018922193 |
| -1.9134171618254476 | -4.619397662556434 |
| -1.2940952255126033 | -4.8296291314453415 |
| -0.6526309611002581 | -4.957224306869052 |
| -9.184850993605148e-16 | -5 |
| 0.6526309611002606 | -4.957224306869052 |
| 1.2940952255126057 | -4.829629131445341 |
| 1.91341716182545 | -4.619397662556433 |
| 2.5000000000000004 | -4.330127018922193 |
| 3.0438071450436026 | -3.9667667014561756 |
| 3.535533905932737 | -3.5355339059327386 |
| 3.9667667014561747 | -3.0438071450436044 |
| 4.330127018922192 | -2.500000000000002 |
| 4.619397662556434 | -1.9134171618254479 |
| 4.8296291314453415 | -1.2940952255126035 |
| 4.957224306869052 | -0.6526309611002584 |
| 5 | -1.2246467991473533e-15 |
| x | y |
|---|---|
| -5 | 0 |
| 5 | 0 |
| x | y |
|---|---|
| 0 | 0 |
| 3.061616997868383e-16 | 5 |
| x | y |
|---|---|
| 4.531538935183249 | 2.1130913087034973 |
| 1.7101007166283442 | 4.698463103929542 |
- Circle
- All points in a plane at a fixed positive distance r from O.
- Chord / diameter
- A chord joins two boundary points; a diameter is a chord through the centre.
The radius goes from centre to boundary. Every radius of the same circle has length r. A diameter contains two radii, so its length is 2r.
The circle is the boundary. The filled-in region is a disc. A point inside has distance less than r; outside has distance greater than r.
Take a moment. What do you think?
PAUSE & TRY · think first, then check with a free account
Question
Is the centre itself on the circle?
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Sign inNew here? Sign up freeNCERT reference: chapter PDF pages 1, 2, 3.
UNDERSTAND · APPLY · CHECK
A diameter is a mirror line
Use symmetry to understand circle properties.A diameter pairs mirror points
y (units)
x (units)
- Circle · Every boundary point is the same distance from its centre.
- Mirror diameter · Reflection axis.
- Equal offsets · Corresponding points have equal distances from the axis.
- PP · (3, 4)
- P′P′ · (3, -4)
P and its mirror both lie on the circle. Equal units on both axes.
Read the plotted values
| x | y |
|---|---|
| 5 | 0 |
| 4.957224306869052 | 0.6526309611002579 |
| 4.8296291314453415 | 1.2940952255126037 |
| 4.619397662556434 | 1.913417161825449 |
| 4.330127018922194 | 2.4999999999999996 |
| 3.9667667014561756 | 3.043807145043603 |
| 3.5355339059327378 | 3.5355339059327373 |
| 3.043807145043603 | 3.9667667014561756 |
| 2.5000000000000004 | 4.330127018922193 |
| 1.9134171618254492 | 4.619397662556434 |
| 1.2940952255126037 | 4.8296291314453415 |
| 0.6526309611002585 | 4.957224306869052 |
| 3.061616997868383e-16 | 5 |
| -0.6526309611002569 | 4.957224306869053 |
| -1.2940952255126041 | 4.8296291314453415 |
| -1.9134171618254485 | 4.619397662556434 |
| -2.499999999999999 | 4.330127018922194 |
| -3.043807145043603 | 3.9667667014561756 |
| -3.5355339059327373 | 3.5355339059327378 |
| -3.9667667014561765 | 3.043807145043602 |
| -4.330127018922194 | 2.4999999999999996 |
| -4.619397662556434 | 1.9134171618254494 |
| -4.829629131445341 | 1.294095225512605 |
| -4.957224306869052 | 0.6526309611002579 |
| -5 | 6.123233995736766e-16 |
| -4.957224306869053 | -0.6526309611002566 |
| -4.829629131445342 | -1.2940952255126017 |
| -4.619397662556435 | -1.913417161825446 |
| -4.330127018922193 | -2.5000000000000004 |
| -3.9667667014561756 | -3.043807145043603 |
| -3.5355339059327386 | -3.5355339059327373 |
| -3.0438071450436044 | -3.9667667014561747 |
| -2.500000000000002 | -4.330127018922193 |
| -1.9134171618254476 | -4.619397662556434 |
| -1.2940952255126033 | -4.8296291314453415 |
| -0.6526309611002581 | -4.957224306869052 |
| -9.184850993605148e-16 | -5 |
| 0.6526309611002606 | -4.957224306869052 |
| 1.2940952255126057 | -4.829629131445341 |
| 1.91341716182545 | -4.619397662556433 |
| 2.5000000000000004 | -4.330127018922193 |
| 3.0438071450436026 | -3.9667667014561756 |
| 3.535533905932737 | -3.5355339059327386 |
| 3.9667667014561747 | -3.0438071450436044 |
| 4.330127018922192 | -2.500000000000002 |
| 4.619397662556434 | -1.9134171618254479 |
| 4.8296291314453415 | -1.2940952255126035 |
| 4.957224306869052 | -0.6526309611002584 |
| 5 | -1.2246467991473533e-15 |
| x | y |
|---|---|
| -5 | 0 |
| 5 | 0 |
| x | y |
|---|---|
| 3 | 4 |
| 3 | -4 |
- Reflection symmetry
- Folding across the line maps the shape onto itself.
- Rotational symmetry of a circle
- Any angle of rotation about its centre leaves the circle unchanged.
Every line through the centre is a reflection axis. A marked point may move when the circle rotates, even though the unmarked boundary looks unchanged.
A diameter is the longest chord. Shorter chords can approach zero length, but two distinct endpoints do not form a zero-length chord.
PAUSE & TRY · think first, then check with a free account
Question
Does a decorated wheel have all the circle’s rotational symmetries?
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Sign inNew here? Sign up freeNCERT reference: chapter PDF pages 3, 4.
UNDERSTAND · APPLY · CHECK
Two points allow many centres
Find where the possible centres must lie.Three of infinitely many circles
y (units)
x (units)
- Centre (0, 0) · Every boundary point is the same distance from its centre.
- Centre (0, 2) · Every boundary point is the same distance from its centre.
- Centre (0, −2) · Every boundary point is the same distance from its centre.
- Perpendicular bisector · All possible centres lie here.
- AA · (-3, 0)
- BB · (3, 0)
A = (−3, 0) and B = (3, 0). All three circles pass through both points. Equal units on both axes.
Read the plotted values
| x | y |
|---|---|
| 3 | 0 |
| 2.974334584121431 | 0.3915785766601547 |
| 2.897777478867205 | 0.7764571353075622 |
| 2.77163859753386 | 1.1480502970952693 |
| 2.598076211353316 | 1.4999999999999998 |
| 2.3800600208737057 | 1.826284287026162 |
| 2.121320343559643 | 2.1213203435596424 |
| 1.826284287026162 | 2.3800600208737057 |
| 1.5000000000000004 | 2.598076211353316 |
| 1.1480502970952695 | 2.77163859753386 |
| 0.7764571353075622 | 2.897777478867205 |
| 0.39157857666015516 | 2.974334584121431 |
| 1.8369701987210297e-16 | 3 |
| -0.39157857666015417 | 2.9743345841214315 |
| -0.7764571353075626 | 2.897777478867205 |
| -1.1480502970952693 | 2.77163859753386 |
| -1.4999999999999993 | 2.598076211353316 |
| -1.826284287026162 | 2.3800600208737057 |
| -2.1213203435596424 | 2.121320343559643 |
| -2.3800600208737057 | 1.8262842870261613 |
| -2.598076211353316 | 1.4999999999999998 |
| -2.77163859753386 | 1.1480502970952697 |
| -2.8977774788672046 | 0.7764571353075631 |
| -2.974334584121431 | 0.3915785766601547 |
| -3 | 3.6739403974420594e-16 |
| -2.9743345841214315 | -0.39157857666015394 |
| -2.897777478867205 | -0.7764571353075611 |
| -2.771638597533861 | -1.1480502970952677 |
| -2.598076211353316 | -1.5000000000000004 |
| -2.3800600208737057 | -1.826284287026162 |
| -2.121320343559643 | -2.1213203435596424 |
| -1.8262842870261626 | -2.380060020873705 |
| -1.5000000000000013 | -2.5980762113533156 |
| -1.1480502970952684 | -2.7716385975338604 |
| -0.7764571353075619 | -2.897777478867205 |
| -0.3915785766601549 | -2.974334584121431 |
| -5.51091059616309e-16 | -3 |
| 0.3915785766601564 | -2.974334584121431 |
| 0.7764571353075633 | -2.8977774788672046 |
| 1.14805029709527 | -2.77163859753386 |
| 1.5000000000000004 | -2.598076211353316 |
| 1.8262842870261617 | -2.3800600208737057 |
| 2.121320343559642 | -2.121320343559643 |
| 2.380060020873705 | -1.8262842870261626 |
| 2.598076211353315 | -1.5000000000000013 |
| 2.7716385975338604 | -1.1480502970952686 |
| 2.897777478867205 | -0.776457135307562 |
| 2.974334584121431 | -0.39157857666015505 |
| 3 | -7.347880794884119e-16 |
| x | y |
|---|---|
| 3.605551275463989 | 2 |
| 3.57470528447856 | 2.470618878840465 |
| 3.482695094980158 | 2.9331853381837654 |
| 3.331095026821144 | 3.3797847376628867 |
| 3.122498999199199 | 3.8027756377319943 |
| 2.860476147980679 | 4.194920546815673 |
| 2.5495097567963927 | 4.549509756796392 |
| 2.1949205468156734 | 4.860476147980679 |
| 1.802775637731995 | 5.122498999199198 |
| 1.379784737662887 | 5.331095026821144 |
| 0.9331853381837654 | 5.4826950949801585 |
| 0.47061887884046527 | 5.57470528447856 |
| 2.2077634143293155e-16 | 5.60555127546399 |
| -0.47061887884046405 | 5.5747052844785605 |
| -0.9331853381837658 | 5.4826950949801585 |
| -1.3797847376628865 | 5.331095026821144 |
| -1.8027756377319937 | 5.122498999199199 |
| -2.1949205468156734 | 4.860476147980679 |
| -2.5495097567963922 | 4.549509756796393 |
| -2.8604761479806795 | 4.1949205468156725 |
| -3.122498999199199 | 3.8027756377319943 |
| -3.331095026821144 | 3.379784737662887 |
| -3.4826950949801576 | 2.9331853381837663 |
| -3.57470528447856 | 2.470618878840465 |
| -3.605551275463989 | 2.0000000000000004 |
| -3.5747052844785605 | 1.529381121159536 |
| -3.482695094980158 | 1.066814661816236 |
| -3.3310950268211448 | 0.6202152623371153 |
| -3.1224989991991987 | 0.197224362268005 |
| -2.860476147980679 | -0.1949205468156734 |
| -2.5495097567963927 | -0.5495097567963922 |
| -2.1949205468156743 | -0.8604761479806782 |
| -1.8027756377319961 | -1.1224989991991983 |
| -1.3797847376628858 | -1.3310950268211443 |
| -0.933185338183765 | -1.482695094980158 |
| -0.47061887884046494 | -1.57470528447856 |
| -6.623290242987946e-16 | -1.6055512754639891 |
| 0.47061887884046677 | -1.57470528447856 |
| 0.9331853381837668 | -1.4826950949801576 |
| 1.3797847376628876 | -1.3310950268211434 |
| 1.802775637731995 | -1.1224989991991987 |
| 2.194920546815673 | -0.8604761479806791 |
| 2.549509756796392 | -0.5495097567963927 |
| 2.860476147980678 | -0.19492054681567428 |
| 3.122498999199198 | 0.19722436226800388 |
| 3.3310950268211443 | 0.6202152623371142 |
| 3.482695094980158 | 1.0668146618162349 |
| 3.57470528447856 | 1.5293811211595347 |
| 3.605551275463989 | 1.9999999999999991 |
| x | y |
|---|---|
| 3.605551275463989 | -2 |
| 3.57470528447856 | -1.5293811211595352 |
| 3.482695094980158 | -1.0668146618162346 |
| 3.331095026821144 | -0.6202152623371133 |
| 3.122498999199199 | -0.19722436226800566 |
| 2.860476147980679 | 0.1949205468156734 |
| 2.5495097567963927 | 0.5495097567963922 |
| 2.1949205468156734 | 0.8604761479806791 |
| 1.802775637731995 | 1.1224989991991987 |
| 1.379784737662887 | 1.331095026821144 |
| 0.9331853381837654 | 1.482695094980158 |
| 0.47061887884046527 | 1.57470528447856 |
| 2.2077634143293155e-16 | 1.6055512754639891 |
| -0.47061887884046405 | 1.5747052844785605 |
| -0.9331853381837658 | 1.482695094980158 |
| -1.3797847376628865 | 1.331095026821144 |
| -1.8027756377319937 | 1.1224989991991992 |
| -2.1949205468156734 | 0.8604761479806791 |
| -2.5495097567963922 | 0.5495097567963927 |
| -2.8604761479806795 | 0.1949205468156725 |
| -3.122498999199199 | -0.19722436226800566 |
| -3.331095026821144 | -0.6202152623371129 |
| -3.4826950949801576 | -1.0668146618162337 |
| -3.57470528447856 | -1.5293811211595352 |
| -3.605551275463989 | -1.9999999999999996 |
| -3.5747052844785605 | -2.470618878840464 |
| -3.482695094980158 | -2.933185338183764 |
| -3.3310950268211448 | -3.3797847376628845 |
| -3.1224989991991987 | -3.802775637731995 |
| -2.860476147980679 | -4.194920546815673 |
| -2.5495097567963927 | -4.549509756796392 |
| -2.1949205468156743 | -4.860476147980679 |
| -1.8027756377319961 | -5.122498999199198 |
| -1.3797847376628858 | -5.331095026821144 |
| -0.933185338183765 | -5.4826950949801585 |
| -0.47061887884046494 | -5.57470528447856 |
| -6.623290242987946e-16 | -5.60555127546399 |
| 0.47061887884046677 | -5.57470528447856 |
| 0.9331853381837668 | -5.482695094980158 |
| 1.3797847376628876 | -5.331095026821144 |
| 1.802775637731995 | -5.122498999199198 |
| 2.194920546815673 | -4.860476147980679 |
| 2.549509756796392 | -4.549509756796393 |
| 2.860476147980678 | -4.194920546815674 |
| 3.122498999199198 | -3.802775637731996 |
| 3.3310950268211443 | -3.379784737662886 |
| 3.482695094980158 | -2.9331853381837654 |
| 3.57470528447856 | -2.4706188788404653 |
| 3.605551275463989 | -2.000000000000001 |
| x | y |
|---|---|
| 0 | -6 |
| 0 | 6 |
- Perpendicular bisector
- A line perpendicular to a segment through its midpoint; its points are equally distant from the endpoints.
Every centre through A and B must satisfy OA = OB. So it lies on the perpendicular bisector of AB. There are infinitely many such centres.
The smallest radius is AB/2, with the centre at the midpoint. Moving farther along the bisector gives larger circles; there is no largest radius.
PAUSE & TRY · think first, then check with a free account
Question
For endpoints 6 units apart, what is the smallest possible radius?
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Sign inNew here? Sign up freeNCERT reference: chapter PDF pages 3, 4, 5.
UNDERSTAND · APPLY · CHECK
Three non-collinear points fix one circle
Construct the circumcentre by intersecting bisectors.A right triangle’s circumcentre
y (units)
x (units)
- Circle · Every boundary point is the same distance from its centre.
- Triangle ABC · AB is a diameter and ∠C is 90°.
- Bisector of AB · Passes through O.
- Bisector of AC · Also passes through O.
- AA · (-3, 0)
- BB · (3, 0)
- CC · (0, 3)
- OO · (0, 0)
This is the right-triangle case, not a drawing of every possible triangle. Equal units on both axes.
Read the plotted values
| x | y |
|---|---|
| 3 | 0 |
| 2.974334584121431 | 0.3915785766601547 |
| 2.897777478867205 | 0.7764571353075622 |
| 2.77163859753386 | 1.1480502970952693 |
| 2.598076211353316 | 1.4999999999999998 |
| 2.3800600208737057 | 1.826284287026162 |
| 2.121320343559643 | 2.1213203435596424 |
| 1.826284287026162 | 2.3800600208737057 |
| 1.5000000000000004 | 2.598076211353316 |
| 1.1480502970952695 | 2.77163859753386 |
| 0.7764571353075622 | 2.897777478867205 |
| 0.39157857666015516 | 2.974334584121431 |
| 1.8369701987210297e-16 | 3 |
| -0.39157857666015417 | 2.9743345841214315 |
| -0.7764571353075626 | 2.897777478867205 |
| -1.1480502970952693 | 2.77163859753386 |
| -1.4999999999999993 | 2.598076211353316 |
| -1.826284287026162 | 2.3800600208737057 |
| -2.1213203435596424 | 2.121320343559643 |
| -2.3800600208737057 | 1.8262842870261613 |
| -2.598076211353316 | 1.4999999999999998 |
| -2.77163859753386 | 1.1480502970952697 |
| -2.8977774788672046 | 0.7764571353075631 |
| -2.974334584121431 | 0.3915785766601547 |
| -3 | 3.6739403974420594e-16 |
| -2.9743345841214315 | -0.39157857666015394 |
| -2.897777478867205 | -0.7764571353075611 |
| -2.771638597533861 | -1.1480502970952677 |
| -2.598076211353316 | -1.5000000000000004 |
| -2.3800600208737057 | -1.826284287026162 |
| -2.121320343559643 | -2.1213203435596424 |
| -1.8262842870261626 | -2.380060020873705 |
| -1.5000000000000013 | -2.5980762113533156 |
| -1.1480502970952684 | -2.7716385975338604 |
| -0.7764571353075619 | -2.897777478867205 |
| -0.3915785766601549 | -2.974334584121431 |
| -5.51091059616309e-16 | -3 |
| 0.3915785766601564 | -2.974334584121431 |
| 0.7764571353075633 | -2.8977774788672046 |
| 1.14805029709527 | -2.77163859753386 |
| 1.5000000000000004 | -2.598076211353316 |
| 1.8262842870261617 | -2.3800600208737057 |
| 2.121320343559642 | -2.121320343559643 |
| 2.380060020873705 | -1.8262842870261626 |
| 2.598076211353315 | -1.5000000000000013 |
| 2.7716385975338604 | -1.1480502970952686 |
| 2.897777478867205 | -0.776457135307562 |
| 2.974334584121431 | -0.39157857666015505 |
| 3 | -7.347880794884119e-16 |
| x | y |
|---|---|
| -3 | 0 |
| 3 | 0 |
| 0 | 3 |
| -3 | 0 |
| x | y |
|---|---|
| 0 | -3.5 |
| 0 | 3.5 |
| x | y |
|---|---|
| -3.5 | 3.5 |
| 3.5 | -3.5 |
- Circumcentre
- The common point of the perpendicular bisectors of a triangle’s sides.
The bisector of AB gives points equally far from A and B. Intersect it with the bisector of AC: the intersection is equally far from all three vertices.
Three distinct collinear points have no common circle. For a triangle, the circumcentre is inside if the triangle is acute, outside if obtuse, and at the hypotenuse midpoint if right-angled.
PAUSE & TRY · think first, then check with a free account
Question
Why is the centre at the midpoint of the hypotenuse in a right triangle?
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Sign inNew here? Sign up freeNCERT reference: chapter PDF pages 5, 6, 7.
UNDERSTAND · APPLY · CHECK
Equal chords make equal central angles
Explain both directions using triangle congruence.The radii supply two of the three sides
y (units)
x (units)
- Circle · Every boundary point is the same distance from its centre.
- Chord AB · Central angle 60°.
- Chord CD · Central angle 60°.
- Radii to A and B · Both length 5.
- Radii to C and D · Both length 5.
- AA · (5, 0)
- BB · (2.5, 4.3301)
- CC · (-4.3301, 2.5)
- DD · (-4.3301, -2.5)
- OO · (0, 0)
Equal central angles in one circle give equal chords. Equal units on both axes.
Read the plotted values
| x | y |
|---|---|
| 5 | 0 |
| 4.957224306869052 | 0.6526309611002579 |
| 4.8296291314453415 | 1.2940952255126037 |
| 4.619397662556434 | 1.913417161825449 |
| 4.330127018922194 | 2.4999999999999996 |
| 3.9667667014561756 | 3.043807145043603 |
| 3.5355339059327378 | 3.5355339059327373 |
| 3.043807145043603 | 3.9667667014561756 |
| 2.5000000000000004 | 4.330127018922193 |
| 1.9134171618254492 | 4.619397662556434 |
| 1.2940952255126037 | 4.8296291314453415 |
| 0.6526309611002585 | 4.957224306869052 |
| 3.061616997868383e-16 | 5 |
| -0.6526309611002569 | 4.957224306869053 |
| -1.2940952255126041 | 4.8296291314453415 |
| -1.9134171618254485 | 4.619397662556434 |
| -2.499999999999999 | 4.330127018922194 |
| -3.043807145043603 | 3.9667667014561756 |
| -3.5355339059327373 | 3.5355339059327378 |
| -3.9667667014561765 | 3.043807145043602 |
| -4.330127018922194 | 2.4999999999999996 |
| -4.619397662556434 | 1.9134171618254494 |
| -4.829629131445341 | 1.294095225512605 |
| -4.957224306869052 | 0.6526309611002579 |
| -5 | 6.123233995736766e-16 |
| -4.957224306869053 | -0.6526309611002566 |
| -4.829629131445342 | -1.2940952255126017 |
| -4.619397662556435 | -1.913417161825446 |
| -4.330127018922193 | -2.5000000000000004 |
| -3.9667667014561756 | -3.043807145043603 |
| -3.5355339059327386 | -3.5355339059327373 |
| -3.0438071450436044 | -3.9667667014561747 |
| -2.500000000000002 | -4.330127018922193 |
| -1.9134171618254476 | -4.619397662556434 |
| -1.2940952255126033 | -4.8296291314453415 |
| -0.6526309611002581 | -4.957224306869052 |
| -9.184850993605148e-16 | -5 |
| 0.6526309611002606 | -4.957224306869052 |
| 1.2940952255126057 | -4.829629131445341 |
| 1.91341716182545 | -4.619397662556433 |
| 2.5000000000000004 | -4.330127018922193 |
| 3.0438071450436026 | -3.9667667014561756 |
| 3.535533905932737 | -3.5355339059327386 |
| 3.9667667014561747 | -3.0438071450436044 |
| 4.330127018922192 | -2.500000000000002 |
| 4.619397662556434 | -1.9134171618254479 |
| 4.8296291314453415 | -1.2940952255126035 |
| 4.957224306869052 | -0.6526309611002584 |
| 5 | -1.2246467991473533e-15 |
| x | y |
|---|---|
| 5 | 0 |
| 2.5000000000000004 | 4.330127018922193 |
| x | y |
|---|---|
| -4.330127018922194 | 2.4999999999999996 |
| -4.330127018922193 | -2.5000000000000004 |
| x | y |
|---|---|
| 5 | 0 |
| 0 | 0 |
| 2.5000000000000004 | 4.330127018922193 |
| x | y |
|---|---|
| -4.330127018922194 | 2.4999999999999996 |
| 0 | 0 |
| -4.330127018922193 | -2.5000000000000004 |
- Angle subtended by a chord
- The angle formed by joining its endpoints to a chosen point.
In the same circle, triangles OAB and OCD have equal pairs of radii. If AB = CD too, SSS congruence makes their central angles equal.
Conversely, equal central angles with the two radius pairs give SAS congruence, so the chords are equal. Comparing chords in circles of different radii would require equal radii as well.
- 1
Join each chord endpoint to O.
- 2
Identify the two equal radius pairs.
- 3
Use the given equal chord for SSS, or given central angle for SAS.
- 4
Read the required corresponding angle or side.
PAUSE & TRY · think first, then check with a free account
Question
What are the three equal sides for the SSS proof?
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Sign inNew here? Sign up freeNCERT reference: chapter PDF pages 7, 8, 9.
UNDERSTAND · APPLY · CHECK
The perpendicular from O halves a chord
Use two right triangles to explain the bisection.A perpendicular makes two right triangles
y (units)
x (units)
- Circle · Every boundary point is the same distance from its centre.
- Chord AB = 12 · Its midpoint is M.
- OM = 8 · OM is perpendicular to AB.
- Radius OA = 10 · The hypotenuse of OMA.
- AA · (-6, 8)
- BB · (6, 8)
- MM · (0, 8)
- OO · (0, 0)
O is the centre. All coordinates use 1 unit = 1 cm. AM = √(10² − 8²) = 6 cm. Equal units on both axes.
Read the plotted values
| x | y |
|---|---|
| 10 | 0 |
| 9.914448613738104 | 1.3052619222005157 |
| 9.659258262890683 | 2.5881904510252074 |
| 9.238795325112868 | 3.826834323650898 |
| 8.660254037844387 | 4.999999999999999 |
| 7.933533402912351 | 6.087614290087206 |
| 7.0710678118654755 | 7.071067811865475 |
| 6.087614290087206 | 7.933533402912351 |
| 5.000000000000001 | 8.660254037844386 |
| 3.8268343236508984 | 9.238795325112868 |
| 2.5881904510252074 | 9.659258262890683 |
| 1.305261922200517 | 9.914448613738104 |
| 6.123233995736766e-16 | 10 |
| -1.3052619222005137 | 9.914448613738106 |
| -2.5881904510252083 | 9.659258262890683 |
| -3.826834323650897 | 9.238795325112868 |
| -4.999999999999998 | 8.660254037844387 |
| -6.087614290087206 | 7.933533402912351 |
| -7.071067811865475 | 7.0710678118654755 |
| -7.933533402912353 | 6.087614290087204 |
| -8.660254037844387 | 4.999999999999999 |
| -9.238795325112868 | 3.826834323650899 |
| -9.659258262890681 | 2.58819045102521 |
| -9.914448613738104 | 1.3052619222005157 |
| -10 | 1.2246467991473533e-15 |
| -9.914448613738106 | -1.3052619222005133 |
| -9.659258262890685 | -2.5881904510252034 |
| -9.23879532511287 | -3.826834323650892 |
| -8.660254037844386 | -5.000000000000001 |
| -7.933533402912351 | -6.087614290087206 |
| -7.071067811865477 | -7.071067811865475 |
| -6.087614290087209 | -7.933533402912349 |
| -5.000000000000004 | -8.660254037844386 |
| -3.8268343236508953 | -9.238795325112868 |
| -2.5881904510252065 | -9.659258262890683 |
| -1.3052619222005162 | -9.914448613738104 |
| -1.8369701987210296e-15 | -10 |
| 1.3052619222005213 | -9.914448613738104 |
| 2.5881904510252114 | -9.659258262890681 |
| 3.8268343236509 | -9.238795325112866 |
| 5.000000000000001 | -8.660254037844386 |
| 6.087614290087205 | -7.933533402912351 |
| 7.071067811865474 | -7.071067811865477 |
| 7.933533402912349 | -6.087614290087209 |
| 8.660254037844384 | -5.000000000000004 |
| 9.238795325112868 | -3.8268343236508957 |
| 9.659258262890683 | -2.588190451025207 |
| 9.914448613738104 | -1.3052619222005168 |
| 10 | -2.4492935982947065e-15 |
| x | y |
|---|---|
| -6 | 8 |
| 6 | 8 |
| x | y |
|---|---|
| 0 | 0 |
| 0 | 8 |
| x | y |
|---|---|
| 0 | 0 |
| -6 | 8 |
- Distance from centre to chord
- The perpendicular length from the centre to the chord.
With OM perpendicular to AB, OMA and OMB are right triangles. OA = OB, and OM is common. RHS congruence gives AM = BM.
Conversely, for a non-diameter chord, the segment joining its midpoint to O is perpendicular to it. For a diameter, the midpoint is O itself; use the perpendicular line through O rather than a zero-length OM.
PAUSE & TRY · think first, then check with a free account
Question
Why is a sloping line from O to an endpoint not the chord’s distance from O?
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Sign inNew here? Sign up freeNCERT reference: chapter PDF pages 9, 10, 11.