How to find the common ratio of a quadrilateral

Question

Please eplain step by step

Answer 1

I imagine you are referencing the common ratio between similar quadrilaterals as you might find on worksheets such as this: www.humbleisd.net/cms/lib2/TX01001414/Centricity/Domain/4888/More%20Similar%20Polygons%20Notes.doc

The idea is much like that found for any instance of similarity where one shape is a magnified version of the other. In such instances, all the angles in both shapes are congruent to each other, but their sides may be different sizes. In fact, in similar shapes, the ratio of any two corresponding sides is a common ratio or factor. If the side on one shape is a and the side on the other shape is b, a common ratio between the two shapes r will exist so that a = b × r. Thus, to find a common ratio simply divide one corresponding side's length by the other: r = a ÷ b. It's as simple as that. This also implies that every other corresponding pair of sides in a similar shape will have the same ratio (r).

 

One thing you should watch out for, however, is the fact that the areas of these shapes will be changed by the common ratio squared. In a square, this is easy to see as a² = (br)² = b² × r². Note that this fact can be proven for every other shape as well!

Answer 2

Hi Ashjah,

Determining the common ratio of a quadrilateral means finding a common factor of its four angles, such that

Ax+Bx+Cx+Dx = 360°,

where ABCD are the ratios of angles relative to each other, and x is the common factor.

 

The best way to show you the steps is to demonstrate them in a sample problem.

 

Let's say the angles of a particular quadrilateral are 60°, 90°, 90°, 120°.  To determine the common ratio, we want to find the lowest common factor.

 

Let's start by factoring out 3:

60/3 = 20

90/3 = 30

90/3 = 30

120/3 = 40

 

Now let's factor out 10:

20/10 = 2

30/10 = 3

30/10 = 3

40/10 = 4

 

Since we can't factor any further, we can determine our lowest common factor to be 3*10=30.

 

This is your common ratio.  You'll also note we've determined the ratio of the angles above, 2:3:3:4.  Thus, you can double-check your work by inserting the ratio of the angles and the common ratio into our original equation:

Ax+Bx+Cx+Dx = 360°

2*30+3*30+3*30+4*30 = 360°

360° = 360°