MPM 2D - A Mathematical Verification (Proof)
3 steps:
Statement: what are the properties that you going to prove in order to verify the claim?
Ie. Verify the properties of a parallelogram ABCD
Prove: opposite slopes are equal
𝑦 −𝑦
Solve: 𝑚𝐴𝐶 = 𝑥2 −𝑥1 , 𝑚𝐵𝐷 =,
2
Conclude:
1
𝑚𝐴𝐵 =,
𝑚𝐴𝐶 = 𝑚𝐵𝐷
𝑎𝑛𝑑
𝑚𝐴𝐵 = 𝑚𝐶𝐷
𝑚𝐶𝐷 =
∴ The shape is a parallelogram because
2 pairs of opposite sides have equal slopes so they are parallel
FOR THESE OTHER QUADRILATERALS:
Rectangle: prove it is a parallelogram PLUS adjacent sides are perpendicular
Use slope equation as above PLUS show that adjacent sides are negative reciprocals
Square: prove it is a rectangle PLUS all sides are equal length
Use slope equation as above PLUS
𝑑𝐴𝐵 = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑠𝑖𝑑𝑒𝑠
Rhombus: prove 2 pairs of opposite sides are parallel PLUS all sides are equal length
FOR TRIANGLE properties:
SCALENE: prove all side lengths are not equal
Use distance equation
𝑑𝐴𝐵 = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑠𝑖𝑑𝑒𝑠
Isosceles: prove 2 sides are equal, one different
Use distance equation
𝑑𝐴𝐵 = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑠𝑖𝑑𝑒𝑠
Equilateral: prove 3 sides are equal
Use distance equation
𝑑𝐴𝐵 = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑠𝑖𝑑𝑒𝑠
Right-angled: prove one 2 slopes are negative reciprocals of each other
Use slope equation 𝑚𝐴𝐵 =
𝑦2 −𝑦1
𝑥2 −𝑥1
Solving Word Problems
4 steps:
Identify what the problem is asking you to solve.
Sketch the situation.
Plan the steps you will need to solve.
Write a concluding statement that answers the question. Pay attention to any units that may be involved.