Proportional Segments
What Are Proportional Segments?
When we say that line segments are proportional, we mean that their lengths share the same ratio. Proportional segments are at the heart of similarity in geometry and appear everywhere — from maps and scale drawings to architecture and art.
Formal Definition: Two pairs of segments are proportional if the ratio of the lengths of one pair equals the ratio of the lengths of the other pair. If segments have lengths $a$, $b$, $c$, and $d$, they are proportional when: $$ \frac{a}{b} = \frac{c}{d} $$This is also written as $a : b = c : d$ and read as "$a$ is to $b$ as $c$ is to $d$."
Key Properties of Proportions
1. Cross-Multiplication Property
If $\frac{a}{b} = \frac{c}{d}$, then:
$a \cdot d = b \cdot c$
The products of the means and extremes are equal.
2. Alternation Property
If $\frac{a}{b} = \frac{c}{d}$, then:
$\frac{a}{c} = \frac{b}{d}$
3. Inversion Property
If $\frac{a}{b} = \frac{c}{d}$, then:
$\frac{b}{a} = \frac{d}{c}$
4. Addition Property
If $\frac{a}{b} = \frac{c}{d}$, then:
$\frac{a + b}{b} = \frac{c + d}{d}$