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# Geometry/Geometry and Scale Factor

Question
I need help understanding a proportion scale factor problem:

A triangle has an area of 20 square in. Every dimension of the triangle is multiplied by a scale factor, and the new triangle has an area of 180 square inches. What is the scale factor?

Please explain how to find the scale factor. Thanks! :)

Hey Rebecca,

In a scale transformation, the scale factor is the ratio of the new side length versus the old side length.  (I say side length, but any one-dimensional length will work, provided you use the corresponding one in both figures.)  For instance, if I have a square with a side length of 5, and I transform it so that the side is now 10, the scale factor will be 2 (because 10/5=2).

When dealing with area or any two-dimensional measurement, the area will change with the scale factor to the power of 2.  So in the previous example, the ratio of the sides is 2, but the ratio of the areas is
(area of new square)/(area of old square)=100/25=4=2².
So the ratio of the areas is the scale factor to the power of 2.
(In the same vein, the ratio of volumes is the scale factor to the power of 3.)

If you are given the two areas, you can use this relation to go backwards and find the scale factor.

Try it, and tell me if you need any more help.
Azeem

Geometry

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#### Azeem Hussain

##### Expertise

I welcome your questions on algebra, 2D and 3D geometry, parabolic functions and conic sections, and any other mathematical queries you may have.

##### Experience

4 years as a drop-in and by-appointment tutor at Champlain College. Private tutor for dozens of clients over the past 8 years.

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CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry

Education/Credentials
Bachelor of Science, Major Mathematics and Major Economics, McGill University, 2014. Diploma of Collegiate Studies; Pure and Applied Science, Champlain College Saint-Lambert, 2010.