# Ch 2_A Mathematical Verification

```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
π¦ −π¦
Solve: ππ΄πΆ = π₯2 −π₯1 , ππ΅π· =,
2
Conclude:
1
ππ΄π΅ =,
ππ΄πΆ = ππ΅π·
πππ
ππ΄π΅ = ππΆπ·
ππΆπ· =
∴ The shape is a parallelogram because
2 pairs of opposite sides have equal slopes so they are parallel
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
ππ΄π΅ = √(π₯2 − π₯1 )2 + (π¦2 − π¦1 )2 πππ πππ π ππππ
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
ππ΄π΅ = √(π₯2 − π₯1 )2 + (π¦2 − π¦1 )2 πππ πππ π ππππ
Isosceles: prove 2 sides are equal, one different
Use distance equation
ππ΄π΅ = √(π₯2 − π₯1 )2 + (π¦2 − π¦1 )2 πππ πππ π ππππ
Equilateral: prove 3 sides are equal
Use distance equation
ππ΄π΅ = √(π₯2 − π₯1 )2 + (π¦2 − π¦1 )2 πππ πππ π ππππ
Right-angled: prove one 2 slopes are negative reciprocals of each other
Use slope equation ππ΄π΅ =
π¦2 −π¦1
π₯2 −π₯1
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.
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