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.