Geometry
Theory
Question and Answer
Questions

Answers


Q01. Diagrams

Top left diagram
  • Pythagorean Law : c^2 = a^2 + b^2
  • Proof : see MD2002 13 09
Top mid diagram
  • Mid point theory of triangle.
  • D is mid point on AB and E is mid point on AC.
  • Then DE = BC/2
Top right diagram
  • Example : Application of mid point theory.
  • Join the mid points of traingle will give 4 triables.
  • These four triangles are congruent.
Bottom left diagram
  • Mid point theory of triangle.
  • D is mid point of AB and E is mid point of AC.
  • Then DE is parallel to BC.
Bottom mid diagram
  • Example : Application of mid point theory.
  • Join the 4 mid points of quadrilateral will give a parallelogram.
Bottom right diagram
  • Let G be the gravity center of a triangle.
  • Let O be the ex-center of a triangle.
  • Let H be the orthocenter center of a triangle.
  • Then GOH will be colinear and HO = 2*OG (OG : HO = 1 : 2)
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Q02. Bisector theory
Bisector of line AB.
  • If PD is a bisector of line AB and D is on line AB.
  • Then PD is perpendicular to AB.
  • Then PA = PB and DA = DB.
Bisector of an angle.
  • Angle BAC. Point A is the common point. AB and AC are the rays.
  • If PA is the bisector of angle BAC.
  • Then angle PAC = angle PAB.
  • Then point P to ray AB = point P to ray AC.
  • Point P to ray AB is the line perpendicular to AB.
Go to Begin

Q03. Mid point theory of triangle
  • Construction
    • Draw triangle ABC.
    • Let D be mid point of AB.
    • Draw line DE parallel to BC.
  • Statement
    • Prove that E is the mid point of AC.
  • Hint
    • Make BDPC as parallelogram.
    • Prove that triangle ADE is congruent to triangle CEP.
  • Proof
    • Draw CP parallel to AB.
    • Produce DE to P.
    • Since CP parallel to BD and HP parallel to BC (By construction).
    • Hence BDPC is a parallelogram.
    • Hence DAC = PCE (alternate angles equal).
    • Angle CEP = angle AED (vertical angle equal).
    • CP = BD = DA. Hence triangle ADE is congruent to triangle CEP.
    • Hence AE = CE and E is the mid point of AC.
Go to Begin

Q04. Mid point theory of triangle
  • Construction
    • Draw triangle ABC.
    • Let D be mid point of AB. Let E be the mid point of AC
  • Statement
    • Prove that DE = BC/2.
  • Hint
    • Since D and E are mid-points, hence DE is parallel to BC.
    • Make BDPC as parallelogram.
    • Prove that triangle ADE is congruent to triangle CEP.
  • Proof
    • Produce DE to P and let DE = EP.
    • Join CP.
    • Since AE = EC and DE = EP, hence ADCP is a parallolgram.
    • Since DAC = PCE (alternate angles equal).
    • Angle CEP = angle AED (vertical angle equal) and AE = EC.
    • Hence triangle ADE is congruent to triangle CEP.
    • Hence DE = DP/2 and DE parallel to BC.
Go to Begin

Q05. Join the mid points of quadrilateral will give a parallelogram.
Construction
  • Draw a quadrilateral ABCD.
  • Let the mid points be E on AB, F on BC, G on CD and H on DA.
Statement
  • Prove that EFGH is a parallelogram.
Proof
  • Draw line AC.
  • In triangle ABC, EF = AC/2 and EF parallel to AC.
  • In triangle ADC, GH = AC/2 and EF parallel to AC.
  • Hence EFGH is a parallelogram.
Go to Begin

Q06. Join the mid poins of a triangle will give 4 congruent triangles

Construction
  • Draw a triangle ABC.
  • Let the mid points be D on AB, E on BC and F on CD.
Statement
  • Prove that the four triangles are congruent.
Proof
  • DF = CE and DE = FC. EF is common side. Hence triangle DEF congruent to CEF.
  • EF = BD and FD = BE. DC is common side. Hence triangle DEF congruent to BDE.
  • EF = AD and DE = AF. DF is common side. Hence triangle DEF congruent to ADF.
Go to Begin

Q07. Answer
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Q08. Answer
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Q09. Answer
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Q10. Answer
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