8.1 Ratios and Proportions
Notes
A ratio compares two numbers, like $3:5$ or $\displaystyle \frac{3}{5}$. A proportion says two ratios are equal.
- Cross products: if $\displaystyle \frac{a}{b} = \frac{c}{d}$, then $ad = bc$.
- An extended ratio like $2:3:4$ can be written $2x$, $3x$, $4x$.
- The geometric mean of $a$ and $b$ is $\displaystyle \sqrt{ab}$.
- Example: the angles of a triangle are in ratio $2:3:4$. Then $9x = 180$, $x = 20$, and the angles are $40^\circ$, $60^\circ$, $80^\circ$.
Practice
Solve $\displaystyle \frac{x}{12} = \frac{5}{8}$.
$8x = 60$, so $x = 7.5$.
The sides of a triangle are in the ratio $3:4:5$ and its perimeter is 60. What is the longest side?
$12x = 60$, so $x = 5$ and the longest side is $5 \cdot 5 = 25$.
What is the geometric mean of 4 and 25?
$\displaystyle \sqrt{4 \cdot 25} = \sqrt{100} = 10$.
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