Ratio, Proportion |
Intro
Suppose you are a graphic artist hired by a company to enlarge their triangular logo to billboard size. You need to create an exact replica of the logo on the letterhead, only much larger, so you use ratio and proportion and similar triangles to create the big logo. Using ruler and protractor, you measure to find the triangle is right-angled at the upper vertex and the sides measure 3 cm, 4 cm and 5 cm. (a common Pythagorean triple). This means that you must make the larger triangle with sides that are are multiples of 3, 4 and 5. Which multiple you choose will depend on the size of the target billboard. If it is large enough, you could make the billboard logo sides
3, 4 and 5 meters long. If not, you could use 30, 40 and 50 centimeters. Whatever size you choose, the ratio of the sides will be 3 : 4 : 5.
This is why we study these topics under the heading of Geometry. A knowledge of ratio, proportion and similar figures is an essential tool used by professionals in their daily work. Every architectural blueprint, city map or commercial design is created using the properties and principles of ratio and proportion. Today, with CAD (computer assisted drawing) software, it is easier than ever to reproduce exact reductions or enlargements of geometric figures with just a few clicks. Psychologists have recently discovered that faces and bodies of specific symmetric proportions tend to be the most attractive to the opposite sex across cultural lines.
Ratio and Proportion
Rational numbers are fractions. The word rational contains the word ratio to tell us we're dealing with fractions. A ratio and a fraction are identical if and only if the ratio is a 2-term ratio; that is, it is made up of just 2 terms -- no more.
Ratios with more than 2 terms are not fractions.
The three-term ratio 6 : 3 : 2 can not be written as a fraction however,
the two term-ratio 2 : 3 can be written as the fraction.
If b ! 0, the ratio a : b is equal to the fraction. |
The ratio 24 : 12 : 8 is equivalent to 6 : 3 : 2 because
24 = 6 (4), 12 = 3 (4) and 8 = 2 (4)
so we can write that 24 : 12 : 8 = 6 : 3 : 2
and since 60 = 6 (10), 30 = 3 (10) and 20 = 2 (10)
we can write that 60 : 30 : 20 = 6 : 3 : 2.
A proportion is a statement of equality between two ratios. |
The statement 60 : 30 : 20 = 6 : 3 : 2 is a proportion.
x : y : z = a : b : c is true if and only if there exists a real number k (k ! 0) such that x = ka, y = kb and z = kc. |
The values x, y, z, a, b, c are called the proportionals.
k is called the proportionality factor.
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Example 1
Are 3, 8 and 10 proportional to 12, 32, and 40? Justify your answer.
If the answer is yes, we must be able to find k (the proportionality factor).
12 = 3 (4) | 32 = 8 (4) | 40 = 10 (4) |
we can see that 3 : 8 : 10 = 12 : 32 : 40 and k = 4.
Therefore the answer is yes, 3, 8 and 10 are proportional to 12, 32 and 40.
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Ratios of Line Segments and Areas
When we speak of the ratio of line segments, we are really referring to the ratio of the lengths of those line segments. So, if we know that AB = 32 cm. and BC = 50 cm, the ratio
AB : BC = the ratio of 32 : 50 or the fraction .
Similarly, if we refer to the ratio of the areas of two triangles with areas of 50 cm^{2} and
66 cm^{2} , we can make the statement
Example 2
The ratio of the sides AB : BC : AC of TRIANGLE ABC is 30 : 40 : 50 = 3 : 4 : 5.
The ratio of the sides DE : EF : DF of TRIANGLE DEF is 6 : 8 : 10 = 3 : 4 : 5.
Thus, AB : BC : AC = DE : EF : DF = 3 : 4 : 5.
So the sides of TRIANGLE ABC are proportional to the sides of TRIANGLE DEF.
Since both triangles are right-angled, we can find their areas using the formula A = ½bh.
For TRIANGLE ABC: A = ½(40)(30) = 600 cm^{2}. For TRIANGLE DEF: A = ½(8)(6) = 24 cm².
Therefore we can state that .
Now let's investigate the relationship between the two ratios: ie. the sides and the areas.In TRIANGLE ABC the ratio of the sides is 30 : 40 : 50. In TRIANGLE DEF it is 6 : 8 : 10. The proportionality factor between these two is 5. Recall that 25 = 5².
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If the sides of two triangles are proportional with a proportionality factor of k, then the areas of the two triangles are also proportional with a proportionality factor of k². |
The converse is also true. Given the ratio of the areas of 2 similar triangles, we can find the ratio of their sides and heights by taking the square root.
Example: The area of triangle ABC is 9 times that of triangle DEF. Find the ratio of their sides.
Solution: The ratio of their areas is 9 : 1, so the ratio of their corresponding sides is 3 : 1.
The units give us a clue. Area is measured in SQUARE units -- whereas sides, base, and height are measured in LINEAR units.
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Proportions Involving Two-term Ratios
In the proportion a : b = c : d, where a, b, c, and d are elements in R, the set of Real numbers,
a and d are called the extremes; (outer terms)
b and c are called the means; (middle terms)
d is called the fourth proportional to a, b and c.
THEOREM: If a, b, c, d ! 0, and , then: 1) ad = bc (cross multiply) 2) (alternate the means) 3) (invert) 4) (add unity) 5) (subtract unity) 6) |
Let's illustrate each property numerically:
Since 1) 1 × 12 = 4 × 3 (cross multiply) 2) (alternate the means) 3) (invert) 4) (add unity) 5) (subtract unity) 6) |
1) says top left times bottom right = top right times bottom left.
2) says switch the middle terms (means).
3) says flip both fractions
4) says add 1 to both sides. (3/3 added on left, 12/12 added on right)
5) says subtract 1 from both sides. (3/3 subtracted on left, 12/12 subtracted on right)
6) says we can add or subtract the numerators and denominators of a proportion without changing the ratio.
Example 3
Find the fourth proportional to 3, 5, and 8.
Let x represent the fourth proportional.
Then (definition of fourth proportional)
So 3x = 40 (cross multiply)
therefore (divide)
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Proportions Involving Three-term Ratios
THEOREM: If with a, b, c ! 0 then: 1) a, b, c are in continued proportion; 2) b is the geometric mean or mean proportional of a, and c; 3) c is the third proportional to a and b. |
Example 4
Find the positive mean proportional of 4 and 64.
Let x = the value we're looking for:
Then (definition)
therefore x^{2} = 256 (cross multiplication)
therefore x = ± 16 (take square root of both sides)
Since we want the positive value, our answer is x = 16.
So
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Practice
1) State the property of proportions applied to turn the first proportion into the second.
a) | b) | c) |
d) | e) | f) |
2) Use the properties of proportion to solve:
a) | b) | c) | d) |
3) The ratio of the areas of 2 triangles is . If the smaller base measures 30 cm,
how long is the larger base?
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Solutions
1) State the property of proportions applied to turn the first proportion into the second.
a) alternate the means |
b) sum of numerators/denomins. |
c) sum of numerators & denomins. |
d) invert fractions |
e) difference of numerators |
f) difference of numerators |
2) Use the properties of proportion to solve:
a) cross multiply: 10a + 15 = 4a |
b) cross multiply 3a + 15 = 7a – 28 |
c) multiply right by 3/3 a² – 4 = 12 |
d) cross multiply 2x² + 7x = 4 + 2x² |
3) The ratio of the areas = , so the ratio of the bases = . If the smaller base measures 30 cm, the larger one is 7/5 of 30 = 42 cm
(plane geometry MathRoom index)
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