Summary Grade 12: Proportionality Geometry

Proportionality
Geometry

, Parallel Line Theorems
Rate and proportion
Ratio & Proportion
RATIO: The relative size of two quantities expressed as the quotient of one divided by the other
PROPORTION: A relationship of equality btwn two ratios. Four quantities are in proportion if:
a and d = extremes
a c b and c = means
=
b d d = fourth proportional
b = second proportional


Proportionality Rule
INVERTENDO:
a x b y
= =
b y a x
ALTERNANDO
a x y x a b
= = =
b y b a x y
COMPONENDO
a x a+b x+y
= =
b y b y
DIVIDENDO
a x a-b x-y
= =
b y b y

Area of triangles
CASES
1. The areas of triangles with equal altitudes are in the same proportion as the length of their bases
B E
△ △
GIVEN: ABC AND DEF
With same h
h
h △
CONCLUSION: ABC = AB
△ DEF DE

A CD F
2. Triangles on the same base (or equal bases) and between the same parallel lines are equal in area
C D B E
△ △
ABC = ABD △
Area ABC = 1/2 x AB x h
△
Area ABC =1/2 x AB x h △ DEF = 1/2 x DE x h
△
Area ABD =1/2 x AB x h BUT AB = DE (equal bases)
∴ △
Area ABD = ABC△ ∴ △ △
Area ABC = DEF

A B A C D F