Mathematics Dictionary
Dr. K. G. Shih
Circle : rule and theory
Subjects
Symbol Defintion
Q00 |
- Outlines
Q01 |
- Tangent and normal
Q02 |
- Two tangents from point p to circle are equal
Q03 |
- Bisector of chord passing center
Q04 |
- Chord and chord relation
Q05 |
- Tangent and secant relation
Q06 |
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Q07 |
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Q08 |
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Q09 |
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Q10 |
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Answers
Q01. Tangent and normal
Definition
Tangent to a circle is a line which only one point meets the circe
Normal is a line perpendicular to tangent and passing the center of the circle
Properties of tangent
Point P on tangent, then PC is perpendicular to tangent. C is center of circle
Go to Begin
Q02. Tangents from point to circle are equal
Constructione
Draw a circle with center C
At point P draw two tangents
Let tangents meet circle at point A and point B
Proof
Right triangle PAC is congruent to right triangle PBC
AC = BC.
Angle PAC = 90 and Angle PBC = 90
PC is common side of two triangles
Hence Right triangle PAC is congruent to right triangle PBC (SSS)
Go to Begin
Q03. Bisector of chord passing center
Construction
Draw a chord AB
Mid point of AB is P
Join P and center C
PC is the bsector of the chord
Proof
Triangle APC is congruent to triangle BPC
AC = BC = radius
PA = PB (construction)
PC is common side of these two triangle
Hence Triangle APC is congruent to triangle BPC
ABP is a line (the chord)
Angle APC + angle BPC = 180
But angle APC = angle BPC
Hence angle APC = angle BPC and AC = BC
Hence PC is bisector of chord AB
Go to Begin
Q04. Chord and chord relation
Construction
Draw chord AB
Draw chord CD
Two chords meet at P
Proof
Angle APC = angle BPD (vertical angles)
Angle ACP = angle PBD (Face same arc AD)
Angle CAP = angle PDB (Face same arc BC)
Hence triangle APC is similar to BPD
AP*PB = CP*PD. This is the chord and chold relation
Go to Begin
Q05. Tangent and secant relation
Construction
Draw tnagent PT and T is on circle
Draw secant PAB. A and B are on circle
Proof : PT*PT = PA*PB
Triangle PTA is congruent to triangle PTB
Angle P is in common for both triangle
Angle PBT = (arc AT)/2
Angle PTA = (arc AT)/2)
Hence angle PBT = angle PTA
Hence 3rd angles of two triangle are also equal
Hence riangle PTA is similar to triangle PTB
Hence PT*PT = PA*PB
Go to Begin
Q06. Secant and secant relation
Construction
Draw secant PAB. A and B are on circle
Draw secant PCD. C and D are on circle
Proof : PA*PB = PC*PD
Prove that triangle PAD is similar to triangle PCB
Angle PDA = angle PBC = (arc AC)/2
Angle P is common angle for triangle PAD and PCB
Since two corresponding angles are same, hence 3rd angle is same.
Hence triangle PAD is similar to triangle PCB
Hence PA*PB = PC*PB
Go to Begin
Q07. Two circles have common tangent
Construction
Draw two circles with same radius contacting at P.
Draw a tangent PQR and passes P. Q on top and R at bottom
Draw line APB. A is on one circle and B on other
Draw line CPD. C is on one circle and D on other
Proof : AC is parallel to BD
Angle PBD = (arc PD)/2
Angle DPQ = (arc PD)/2
Angle CAP = (arc PC)/2
Angle CPR = (arc PC)/2
Since DPQ = CPR (vertical angle)
Hence angle B = angle A
Hence AC is parallel to BD (Alternate angle equal)
Go to Begin
Q08. Answer
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Q09. Answer
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Q10. Answer
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