PROBLEM SOLVING IN GEOMETRY WITH PROPORTIONS

Properties of Proportions

1. Cross Product Property :

The product of the extremes equals the product of the means.

If a/b  =  c/d, then ad  =  bc.

2. Reciprocal Property :

If two ratios are equal, then their reciprocals are also equal.

If a/b  =  c/d, then b/a  =  d/c.

3. Alternendo Property :

If a/b  =  c/d, then a/c  =  b/d.

4. Componendo Property :

If a/b  =  c/d, then (a + b) / b  =  (c + d) / d.

Using Properties of Proportions

Example 1 :

Say, whether the statement given below is true.

If x / 4  =  y / 16, then x / y  =  1 /  4

Solution :

Given :

x / 4  =  y / 16

Using Alternendo Property,

x / y  =  4 / 16

Simplify.

x / y  =  1 / 4

Hence, the statement is true.

Example 2 :

Say, whether the statement given below is true.

If p / 3  =  q / 5, then (p + 3) / 3  =  (q + 3) /  5

Solution :

Given :

p / 3  =  q / 5

Using Componendo Property,

(p + 3) / 3  =  (q + 5) / 5

Because (p + 3) / 3    (q + 3) /  5, the conclusions are equivalent.

Hence, the statement is false.

Example 3 :

In the diagram shown below,

PQ / QS  =  PR / RT

Find the length of QS.

Solution :

Given :

PQ / QS  =  PR / RT

Substitute.

16 / x  =  (30 - 10) / 10

Simplify.

16 / x  =  20 / 10

16 / x  =  2 / 1

By reciprocal property, we have

x / 16  =  1 / 2

Multiply each side by 16.

16 ⋅ (x / 16)  =  (1 / 2) ⋅ 16

Simplify.

x  =  8

Hence, the length of BD is 8 units.

Geometric Mean

The geometric mean of two positive numbers a and b is the positive number x such that

a / x  =  x / b

If we solve this proportion for x, we find that

x  =  √(a ⋅ b)

which is a positive number.

Fro example, the geometric mean of 8 and 18 is 12.

Because 8 /12  =  12 / 18 and also because

√(8 ⋅ 18)  =  √144  =  12

Using Geometric Mean

Example :

International standard paper sizes are commonly used all over the world. The various sizes all have the same width-to-length ratios. Two sizes of paper are shown below, called A4 and A3. The distance labeled x is the geometric mean of 210 mm and 420 mm. Find the value of x.

Solution :

Using the given information, write proportion.

210 / x  =  x / 420

Using cross product property,

x2  =  210 ⋅ 420

x2  =  √(210 ⋅ 420)

Simplify.

x  =  √(210 ⋅ 210 ⋅ 2)

x  =  210√2

The geometric mean of 210 and 420 is 2102, or about 297.

Hence, the distance labeled x in the diagram is about 297 mm.

Using Proportions in Real Life

Example :

A scale model of the Titanic is 107.5 inches long and 11.25 inches wide. The Titanic itself was 882.75 feet long. How wide was it ?

Solution :

One way to solve this problem is to set up a proportion that compares the measurements of the Titanic to the measurements of the scale model

Problem Solving Strategy :

Multiply each side by 11.25

11.25 ⋅ (x / 11.25)  =  (882.75 / 107.5) ⋅ 11.25

Simplify.

x  =  (882.75 ⋅ 11.25) / 107.5

Using calculator, we have

x  ≈  92.4

Hence, the Titanic was about 92.4 feet wide.

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