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Mathematics Dictionary
Dr. K. G. Shih
Diagram Constructions
Subjects

Answers


Q01. How to bisect an angle ?
View diagram for bisecting an angle
  • Bisector of an angle is the line to bisect the angle.
  • Point on the angle bisector has same distance from the two arms.
    • Let common point of angle be A.
    • Draw a point P on bisector.
    • Draw PB perpendicular to one arm.
    • Draw PC perpendicular to other arm.
    • Prove PB = PC.
      • Angle PAB = angle PAC by construction.
      • Line PA is common line for triangle PAB and triangle PAC.
      • Angle PBA = angle PCA = 90.
      • Hence triangle PBA is congruent to triangle PCA.
      • Hnece PB = PC.
Go to Begin

Q02. How to draw a line parallel to a given line ?
Construction
  • Let given line be AB.
  • Draw a line cross line AB at Q.
  • Draw a point P on the cross line.
  • At point P draw an angle QPC with cross line PQ and let the angle QPC = PQB.
  • The line PC is the line which is parallel to line AB
Proof
  • Angle QPC and PQB are alternative angle and they are equal.
  • Hence PC is parallel to AB.
Conditions of two lines parallel : satisfying following conditions.
  • Condition 1 : The alternate angles are equal.
  • Condition 2 : The corresponding angles are equal.
  • Condition 3 : The two interior angles are supplimentary angles.
View various angles
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Q03. Change a quadrilateral to a triangle with area keeping the same.
Construction
  • Draw a quadrilateral ABCD.
  • Draw a line DP parallel to line AC.
  • Extend line BC and meet line DP at point E.
  • Join A and E. Then triange ABE has same area of quadrilateral ABCD.
Proof
  • Area quadrilateral ABCD = triangle ABC + triangle ACD.
  • Area triangle ABE = triangle ABC + triangle ACE.
  • Triangle ACD and ACE have same base AC.
  • Triangle ACD and ACE have equal height because AB parallel to DE.
  • Hence area of ACD = area of ACE.
  • Hence area of quadrilateral ABCD = area of triangle ABE.
Go to Begin

Q04. Change a pentagon to a triangle with area keeping the same.
  • Similar as Q03, changle pentagon to a quadrilateral.
  • Then changle quadrilateral to a triangle.
Go to Begin

Q05. Draw a circle passing three given points
Construction
  • Let the three points be A, B and C.
  • Make triangle ABC.
  • Bisect line AB and line BC.
  • The bisectors meet at one point which is called ex-center E.
  • Use EA as radius and E as center to draw a circle.
Study : What is Ex-center ? What is the ex-center theory ? Go to Begin

Q06. Two Fixed point A and B. Find locus of P if angle APB keeps constant

  • The locus of P is the arc of circle passing fixed points A and B.
  • This is an application of ex-circle theory.
  • Theory
    • points A and B fixed. Angle APB is constant.
    • Locus of P is arc passing fixed points A and B.
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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Copyright © Dr. K. G. Shih, Nova Scotia, Canada.
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