Fundamental Theorems on Proportionals

• Proportion


  An equation that shows two ratios are equal
• Proportion


  • Can be written in column form or fraction form
  • Denominator (b) should not be equal to zero
  • Denominator (d) should not be equal to zero
• Terms of a proportion


  a, b, c, d
• Means


  b and c
• Extremes


  a and d
• Checking if a pair of ratios form a proportion

  1. Product of the means is equal to the product of the extremes

  2. Cross multiplication (ad = bc)
• Solving for a missing value in a proportion

  1. Cross multiplication (ad = bc)

  2. Dividing both sides
• Similar triangles


  • Corresponding angles are congruent
  • Corresponding sides are proportional
• Deriving proportions from similar triangles

  Using the basic proportionality theorem
• Finding a missing length in a triangle

  1. Applying the basic proportionality theorem

  2. Using properties of proportions (multiplication, inverse, reciprocal, addition, subtraction)
• Multiplication


  1. Multiply e times 15

  2. Multiply 12 times 10
• Multiplying the means and the extremes

  15 times de is equal to 12 times 10
• Solving the equation


  1. Divide both sides by 15

  2. de is 120 divided by 15, which is 8
• There are two ways of solving the equation
• 12 over 8 is equal to 15 over 10


  They are proportions
• Proportions


  When the simplest form of the fractions are equivalent, they are proportions
• The answer is correct
• Triangle ABC


  • AD is 5 cm
  • BD is x + 6 cm
  • AE is 3 cm
  • EC is x + 3 cm
• Finding the value of x

  1. Set up a proportion: AD/DB = AE/EC

  2. Solve the equation to find x = 1.5
• Triangle ACD


  • DB is 3x + 1
  • BC is 3x - 1
  • AE is 5x - 1
  • EC is 4x + 1
• Finding the value of x

  1. Set up a proportion: DB/BC = AE/EC

  2. Solve the equation to find x = 5
• The value of x cannot be 0 as it would result in negative measurements
• Triangle QRS


  • RD is 3 cm
  • DQ is 5 cm
  • RS is 12 cm
• Finding the value of RE

  1. Set up a proportion: RD/RQ = RE/RS

  2. RE is 9/2 cm
• Trapezium ABCD


  • AB is parallel to DC
  • P and Q are points on AB and DC
  • PD is 18 cm
  • BQ is 35 cm
• Finding the value of PA

  1. Set up a proportion: DP/PA = CQ/QB

  2. PA is 42 cm
• The triangles are similar, so their corresponding angles are congruent and their corresponding sides are proportional