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Mathematics Dictionary
Dr. K. G. Shih
Senior High Mathematics
Study Tips and Home Works
How to do the home work ?

  • 1. Answr the questions first in each home work.
  • 2. Then read the related keywords of the given title in SM.
  • 3. Do the questions again as home work.


  • S21 | - 21. Slope and area of triangle
  • S22 | - 22. Find equation of directrix of parabola
  • S23 | - 23. Ex-centers and ex-central triangle
  • S24 | - 24. Pentagon : Change it to an equal area triangle
  • S25 | - 25. Normal distribution
  • S26 | - 26. Area between curve of y = 1/(1 + x^2) and x-axis
  • S27 | - 27. Hyperbola : Compare x^2 - y^2 = -1 and x*y =1
  • S28 | - 28. Quadratic function : Compare y = (x^2)/2 and y = -(x^2)/2
  • S29 | - 29. Quadratic function : y = x^2 - 6*x + 8 and its inverse
  • S30 | - 30. Sketch y = (x^3 + 1)/x
  • S31 | - 31. Induction method
  • S32 | - 32. Convert x*y = 1 to hyperbolic standard form
  • S33 | - 33. Study y = (x^2)/2 - 1
  • S34 | - 34. Stduy y = 1/x
  • S35 | - 35. Divide one side of equilateral triangle into 3 equal parts
  • S36 | - 36. Construct parabola using definition of Locus
  • S37 | - 37. Construct ellipse using definition of Locus
  • S38 | - 38. Prove cos(A+B) = cos(A)*cos(B) + sin(A)*sin(B)
  • S39 | - 39. Equilateral triangle : Sum of distance of inside point to sides
  • S40 | - 40. Points A(7,4), B(3,1) and C(0,k). Find k is AC + BC = Minimum
    • Home Works


    S21. Home Work 21 : Slope and area of triangle

    Construction
    • Draw OXY coordinate
    • Draw point A(0,y) and C(0,u) on y-axis. Let y GT u
    • Draw point B(x,0) and D(v,0) on x-axis. Let v GT x
    • Join AB and CD. Let AB and CD meet at E(4,6)
    Question
    • Area of triangle AOB = 54 and Area of traingle = 48. O is the origin
    • Find area of triangle EFD where F is a point at (4,0)
    Solution : See AN 11 08.
    Go to Begin

    S22. Home Work 2 : Find equation of directrix of parabola

    Question
    • Parabola : y = x^2 - 6*x + 8
    • Find coordinate of vertex and focus
    • Find equation of the drectrix
    Reference
     Go to Begin

    S23. Home Work 23 : Ex-centers and ex-central triangle

    Constructions : Ex-center between AB and AC
    • Draw a triangle ABC
    • Draw bisector of angle A
    • Draw bisector of external angle of angle B
    • Draw bisector of external angle of angle C
    • The bisectors meet at one point J which is ex-center between AB and AC
    • Similary
      • Ex-center K : Between BC and BA
      • Ex-center L : Between CA and CB
    Questions : Triangle JKL is ex-central triangle
    • 1. Prove that
      • J,C,K are colinear
      • K,A,L are colinear
      • L,B,J are colinear
    • 2. Prove that
      • LC is perpendicular to JK
      • JA is perpendicular to KL
      • KB is perpendicular to LJ
    • 3. Prove that triangle ABC is pedal triangle of tiangle JKL
    Method : Use one of following method
    • Construction and measurement
    • Geometrical emthod
    • Coordinate geometrical emthod
    References
    Go to Begin

    S24. Home Work 24 : Pentagon : Change it to an equal area triangle

    Question
    • 1. Draw a pentagon
    • 2. Change the pentagon to an equal area quadrilateral (See GE 05 06)
    • 3. Change the quadrilateral to an equal area triangle (See GE 06 11)

    Go to Begin

    S25. Home Work 25 : Normal distribution

    Question
    • How to find standard normal distrubtion
      • Using table
      • Using computer
    • A student mean grade is 80 with standard devation of 10
      • Find probability to have mark over 90
      • Find probability to have mark below 70
    • Change to standard
      • Mean u = 80 and standard deviation d = 10
      • Standart normal distribution z = (x - u)/d
      • In this question x = 90 or 70
    Text Reference     Computer calculation
    • Study subjects : Program 01
    • Questions
      • Area based on 10 trapzoids. What is the accuracy ?
      • Area based on 100 trapzoids. What is the accuracy ?

    Go to Begin

    S26. Home Work 26 : Area between curve of y = 1/(1 + x^2) and x-axis

    Question
    • Find area from x1 = 0 to x2 = 1 using trapzoid method
      • Use two trapzoids : h = (x2 - x1)/2
      • Find area of two trapzoids
      • Requred area = Sum of the area of two trapzoids
      • Apprximate pi = 4*Required area
    • Find area from x1 = 0 to x2 = 1 using four trapzoids (h = 0.25)
    Text Reference     Computer calculation
    • Study subjects : Program 01
    • Questions
      • Area based on 10 trapzoids. What is the accuracy ?
      • Area based on 100 trapzoids. What is the accuracy ?

    Go to Begin

    S27. Hyperbola : Compare x^2 - y^2 = -1 and x*y =1

    Reference
    Go to Begin

    S28. Quadratic functions : Compare y = (x^2/2) and y = -(x^2)/2

    Reference     Also compare x = (y^2/2) and x = -(y^2)/2
    • See AN 21 : Program 01 05

    Go to Begin

    S29. Quadratic function : Intersection of y = x^2 - 6*x + 8 and its inverse

    Question : How many intersections of y = x^2 - 6*x + 8 and its inverse ?
    • 1. Estimate solutions from graph
    • 2. Find exact solutions
    Reference
    • See GC 13 03

    Go to Begin

    S30. Rational function : Sketch y = (x^3 + 1)/x

    Questions
    • 1. Find the asymptotes
    • 2. Find extreme points from the graph
    Reference
    • See GC 13 05

    Go to Begin

    S31. Induction method

    Prove that
    • 1. Sum[n^2] = n*(n+1)*(2*n+1)/6
    • 2. Sum[1/(n*(n+1)] = 1 - 1/(n+1)
    Reference
    • See AL 14 00

    Go to Begin

    S32. Convert x*y to hyperbolic standard form

    Questions
    • 1. Find the semi-axes and the focal length
    • 2. Find the principal axis in oxy system after rotation
    Reference
    • See AN 14 04

    Go to Begin

    S33. Study y = (x^2)/2 - 1

    Questions
    • 1. Properties of the curve
    • 2. Properties of the parabola
    Reference
    Go to Begin

    S34 : Study y = 1/x

    Questions
    • 1. Properties of the curve
    • 2. Properties of the parabola
    Reference
    Go to Begin

    S35 : Divide one side of equilateral triangle into 3 equal parts

    Construction
    • 1. Draw an equilateral triangle ABC
    • 2. Use the base BC as diameter to draw a semi-circle under BC
    • 3. Divide the semi-cricle arc into 3 equal parts : arc BD = DE = EC
    • 4. Join AD and cut BC at F
    • 5. Join AE and cut BC at G
    • 6. Then BF = FG = GC. Prove it
    Reference
    Go to Begin

    S36 : Construct parabola using definition of locus

    Definition
    • The point P has same distance to fixed point F and fixed line
    • Fixed point is focus and fixed line is directrix
    Reference
    Go to Begin

    S37 : Construct ellipse using definition of locus

    Definition
    • The sum of point P to two fixed point G and F keeping PF + PG = 2*a
    • Where a is constant and is defined as major semi-axis
    • Point F anf G are foci
    Reference
    Go to Begin

    S38 : Prove cos(A+B) = cos(A)*cos(B) + sin(A)*sin(B)

    Construction
    • Draw a circle with center O and OX is x-axis
    • Draw a point A on circle and let angle XOA = A
    • Draw a point B on circle and let angle XOB = B
    • Then angle AOB = A - B and assume angle A GT B
    Method
    • Use distance formula to find distance AB
    • Use cosine law to find AB
    Reference
    Go to Begin

    S39 : Equilateral triangle : Sum of distance of inside point to sides equal height

    Construction
    • Draw an equilateral triangle ABC and AH is the height
    • Point D is inside the triangle
    • Distance to the sides are DP, DQ, DS
    • Prove that DP + DQ + DS = AH
    Reference
    Go to Begin

    S40 : Points A(7,4), B(3,1) and C(0,k). Find k is AC + BC = Minimum

    Construction
    • Draw oxy coordinate
    • Put the points on the oxy plane
    Questions : Find k, if
    • 1. AC + BC is minimum
    • 2. AC^2 + BC^2 is minimum
    Reference
    Go to Begin
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    Copyright © Dr. K. G. Shih, Nova Scotia, Canada.
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