and
Proportions
Study Guides
Big Picture The use of ratios and proportions comes in handy when finding the relationship between two geometric shapes. Ratios and proportions are often an important part of certain properties between two shapes, such as two similar polygons.
Key Terms Ratio: A way to compare two numbers.
Geometry
Ratios
Extended Ratio: A way to compare three or more numbers. Proportion: An equation that has two ratios equal to each other. Scale: A ratio that relates the dimensions in a drawing to the actual dimensions of an object.
Using Ratios and Proportions Scale
A ratio can be written in three ways (y ≠ 0):
Ratios and proportions are used in maps and scale drawings.
1. 2. x:y
• In a scale drawing, the drawing has the same shape
3. x to y
• Ratios are usually written in simplest form. • It is easier to simplify ratios in fraction form. • Ratios with the same simplified form are equivalent. • Example: 7:14 and 1:2 are equivalent ratios. • Simplified ratios should not have units. • The numerator and denominator must have the
but different size as the object it represents.
• Similarly,
a map accurately represents real-life distances at a more convenient size.
Maps and scale drawings typically have a scale that relates the dimensions in a drawing to the actual dimensions.
•
same units before simplifying - multiply by a conversion factor if needed.
• A
scale can have units (e.g. 1 cm to 1 km) or be simplified to not have units
• Example of conversion factor:
(e.g.
An extended ratio of three numbers is written as x:y:z.
The order of the numbers in a ratio matter!
A proportion relates two equivalent ratios:
,
Scale = Drawing dimension:actual dimension
Using the scale, a proportion can be set up to find the actual dimensions.
b and d ≠ 0
• Actual dimension = Scale • drawing dimension
There are four parts to the proportion (a, b, c, d): • b, c are called the means
•
).
Never assume diagrams are drawn to scale - rely on measurements and markings in the diagram.
a, d are called the extremes
Cross-Multiplication Cross-Multiplication Theorem: If
, where a, b, c, and d are real numbers and b and d are not equal to 0, then
the product of the extremes equals the product of the means: ad = bc.
Corollaries • Swapping the means: If a, b, c, and d are nonzero and
, then
• Swapping the extremes: If a, b, c, and d are nonzero and
, then
. .
• Taking the reciprocal (flipping it upside down): If a, b, c, and d are nonzero and • If a, b, c, and d are nonzero and
, then
.
• If a, b, c, and d are nonzero and
, then
.
This guide was created by Nicole Crawford, Jane Li, Amy Shen, and Zachary Wilson. To learn more about the student authors, visit http://www.ck12.org/ about/about-us/team/interns.
, then
.
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Disclaimer: this study guide was not created to replace your textbook and is for classroom or individual use only.
The following manipulations of a proportion will still make it true: