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Mathematics Dictionary
Dr. K. G. Shih
Triangle : Rule and Theory
Subjects


  • GE 07 01 | - Outline
  • GE 07 01 | - Introduction diagrams and definition
  • GE 07 02 | - Rules of congruent of two triangle
  • GE 07 03 | - Rules of similarity of two triangle
  • GE 07 04 | - Mid-point theory of a triangle
  • GE 07 05 | - A triangle can make 4 congruent traingles
  • GE 07 06 | - Angle bisector theory
  • GE 07 07 | - Line bisector theory
  • GE 07 08 | - In-center theory of triangle
  • GE 07 09 | - Circum-center theory of triangle
  • GE 07 10 | - Centroid theory of triangle
  • GE 07 11 | - Ex-center theory of triangle
  • GE 07 12 | - Ortho-center theory of triangle
    • Answers


    GE 07 01. Rule and theory

    Diagrams : Rules and Theory
    Go to Begin

    GE 07 02. Rules of congruent of two triangle

    Rules
    • 1. SSS : Three sides of two triangles have same sides
    • 2. SAS : One angle and its 2 rays are same
    • 3. ASA : Two angles have same common side
    • 4. AAS : It is same as ASA because the 3rd angle of two triangle is also same
    • 5. SSA : Not a rule because it may have two triangles
      • Triangle ABC and UVW
      • AB = UV, AC = UW and angle B = angle V
      • Triangle ABD and UVW
      • AB = UV, AD = UW and angle B = angle V

    Go to Begin

    GE 07 03. Rules of similarity of two triangle

    Rules
    • 1. AAA : Three angles of two triangles are identical
    • 2. ASA : It is same as AAA
    • 3. AAS : It is same as AAA
    • 4. SAS : The ratio of the corresponding sides are equal
    • 5. SSS : It is not a rule
    • 6. SSA : It is not a rule
    Theory 1
    • If triangle ABC is similar to triangle UVW, then
    • AB/UV = BC/VW = CA/WU = r
    • Ratio of corresponding heights is also equal r.
    Theory 2
    • If triangle ABC is similar to triangle UVW, then
    • (Area of riangle ABC)/(area of triangle UVW) = r^2
      • Area of triangle ABC = AB*(Height CD)/2
      • Area of triangle UVW = UV*(Height WP)/2 = (r*AB)*(r*CD)/2 = (r^2)*(AB*CD/2
      • Hence (Area of riangle ABC)/(area of triangle UVW) = r^2
      • Where r is the ratio of the corresponding sides

    Go to Begin

    GE 07 04. Mid points theory of triangle

    1. Join mid points of two sides which is parallel to other side
    • Construction
      • D is mid point of BC, E is mid point of FA and F is mid point of AB
      • Then EF is parallel to AB
    • Proof
      • Let D be mid-point of BC and F be mid-point of AC
      • At point B draw a line BP parallel to AC and let BP = AF = AC/2
      • Hence ABPF is parallelogram (Opposite sides parallel and equal)
      • Hence FP is parallel to AB.
      • Join CP and FB.
      • CFBP is parallelogram (BP = FC and BP parallel to FC)
      • Hence CB and FP bisect each other equally at D.
      • D is the mid point of BC.
      • Hence F and D are mid-point and FD is parallel to AB
    2. Join mid points of two sides which is equal half of the other side
    • Construction
      • D is mid point of BC, E is mid point of FA and F is mid point of AB
      • Then EF = AB/2
    • Proof
      • Since FABP is parallelogram, hence FP = AB
      • Since CFBP is parallelogram, CB and FP bisect each other equally at D
      • Hence FD = DP = AB/2

    Go to Begin

    GE 07 05. Triangle can make 4 equal triangles

    Construction
    • Let D be mid point of BC, E be mid-point of CA and F be mid-point of AB
    • Join D, E and F will have 4 triangle
    Prove that Triangle CDF = DEF = EAF = DFB
    • 1. Triangle CED = Triangle EAF
      • ED = AF and ED parallel to AF
      • Hence angleCED = angle EAF
      • Since EF parallel to BC and angle AFE = FBD = EDC
      • Hence Triangle CED = Triangle EAF based on ASA rule
    • 2. Triangle CED = Triangle DFB
      • DE = FB
      • Angle CDE = angle DBF
      • Angle CED = angle EAF = angle DFB
      • Hence Triangle CED = Triangle DFB (ASA)
    • 3. Triangle EAF = Triangle DEF
      • Angle EAF = angle EDF (Opposite angles of paralellogram)
      • ED = AF (Mid-point theory)
      • Angle DEF = angle AFE
      • Hence triangle EAF = triangle DEF (ASA)

    Go to Begin

    GE 07 06. Angle bisector theory
    Construction
    • Draw angle CAB
    • Draw angle bisector AP
    • Draw PQ perpendicular to AB
    • Draw PR perpendicular to AC
    Proof : PQ = PR
    • Angle PAB = angle PAC (construction)
    • Side AP is common for right triangle PAB and PAC
    • Hence triangle PAQ is congruent to PAR
    • Hence PQ = PR
    • Property of angle bisector : Point on bisector has same distance from rays
    Diagram
    Go to Begin

    GE 07 07. Line bisector theory
    Construction
    • Draw line AB
    • Draw line PC perpendicular to AB and C is mid point of AB
    Proof : PA = PB
    • AC = BC (construction)
    • Angle PCA = angle PCB = 90 degrees (construction)
    • PC is common for triangle PCA and PCB
    • Hence triangle PCA is congruent to triangle PCB
    • Hence PA = PB
    • Property of angle bisector : Point on bisector has same distance from ends
    Diagram
    Go to Begin

    GE 07 08. In-center theory of triangle
    In-center
    • Bisectors of triangles are concurrent to a point I which is called in-center
    Properties of in-center and in-cricle     Diagram
    Go to Begin

    GE 07 09. circum-center theory of triangle
    circum-center
    • Bisectors of sides are concurrent to a point I which is called circum-center
    Properties of circum-center and circum-cricle     Diagram
    Go to Begin

    GE 07 10. Centroid theory of triangle
    Centroid or gravity-center
    • Medians of triangle are concurrent to a point I which is called centroid
    Properties of centroid     Diagram
    Go to Begin

    GE 07 11. Ex-center theory of triangle
    Ex-center
    • Bisector of internal angle A
    • Bisector of external angle B
    • Bisector of external angle C
    • Three bisectors are concurrent to a point J which is called ex-center
    • A triangle has three ex-center : J, K, L
    Properties of ex-center     Diagram
    Go to Begin

    GE 07 12. Ortho-center theory of triangle

    Ortho-center
    • Three altitudes od triangle are concurrent to a point O and is called ortho-center
    Properties of ortho-center
    • Study | Ortho-center theory of triangle
    • Study | TR 17 Pedal triangle and TR 19 ex-central triangle
    Diagram
    Go to Begin

    GE 07 00. Outlines

    Triangle
    • Congruent rules of two triangles
    • Similar rules of two triangles
    • Mid points rules triangle
    Bisectors
    • Angle bisector theory
    • Line bisector theory
    Five centers of triangle
    • Centroid theory of triangle
    • Circum-center theory of triangle
    • Ex-center theory of triangle
    • In-center theory of triangle
    • Ortho-center theory of triangle

    Go to Begin
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