Triangle Geometry - Proportionality
Important concepts:
1. Area of a triangle
1. a. The height or altitude of a triangle is always relative to the chosen base.
1. In all cases, the area of the triangles can be calculated by the formula:
1
1. Area ∆ ABC = 2 x base x height
1. b. Two triangles which share a common vertex have a common height.
1. c. Triangles with equal or common bases lying between parallel lines have the same area.
Area ∆ ABC = Area ∆ DBC Area ∆ EFH = Area ∆ HGE
, 2. Ratios
21 cm
Example:
A B
4 parts C 3 parts
AC : CB = 4 : 3
4 + 3 ≠ 21
Let each part = k
∴ 4k + 3k = 21
k = 3 cm
AC = 12 cm
CB = 9 cm
3. Cross multiplication techniques
𝑎 𝑐
If = , then:
𝑏 𝑑
𝑏 𝑑
𝑎
=𝑐 Invert both fractions
𝑎 𝑏
𝑐
=𝑑 Interchange the one fractions denominator with the others numerator
𝑑 𝑐
𝑏
=𝑎 Interchange the one fractions denominator with the others numerator
ad = bc Multiply both denominators out
𝑎𝑑
𝑏
=c Multiply the one denominator out
𝑏𝑐
𝑑
=a Multiply the one denominator out
Important concepts:
1. Area of a triangle
1. a. The height or altitude of a triangle is always relative to the chosen base.
1. In all cases, the area of the triangles can be calculated by the formula:
1
1. Area ∆ ABC = 2 x base x height
1. b. Two triangles which share a common vertex have a common height.
1. c. Triangles with equal or common bases lying between parallel lines have the same area.
Area ∆ ABC = Area ∆ DBC Area ∆ EFH = Area ∆ HGE
, 2. Ratios
21 cm
Example:
A B
4 parts C 3 parts
AC : CB = 4 : 3
4 + 3 ≠ 21
Let each part = k
∴ 4k + 3k = 21
k = 3 cm
AC = 12 cm
CB = 9 cm
3. Cross multiplication techniques
𝑎 𝑐
If = , then:
𝑏 𝑑
𝑏 𝑑
𝑎
=𝑐 Invert both fractions
𝑎 𝑏
𝑐
=𝑑 Interchange the one fractions denominator with the others numerator
𝑑 𝑐
𝑏
=𝑎 Interchange the one fractions denominator with the others numerator
ad = bc Multiply both denominators out
𝑎𝑑
𝑏
=c Multiply the one denominator out
𝑏𝑐
𝑑
=a Multiply the one denominator out
