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# Scale Factor

If you like model airplanes, you probably already know all about scale factor. If we have a kit to build a $\frac{1}{72}$ Jetstream 200, for example, this means that we're building a model plane that is exactly ${\frac{1}{72}}_{}\text{nd}$ the size of a real Jetstream 200 across all 3 dimensions. The same general concept can be applied to geometric figures.

## What is scale factor?

If we have two similar figures of different sizes, their scale factor tells us the ratio of one size to the other. For example, we might have these two triangles:

As we can see, these two triangles are similar. We can also see that the larger triangle has a side that is four times the size of the corresponding side of the smaller triangle. we can solve for the scale factor by finding the ratio of two corresponding sides.

$\frac{\mathrm{XY}}{\mathrm{UV}}=\frac{12}{3}=4$

Because these two triangles are similar, this means that the scale factor is "4."

Which then tells us the remaining two unknown sides of the larger triangle

$\frac{\mathrm{ZY}}{\mathrm{WV}}=\mathrm{WV}×4=4×4=16$

$\mathrm{ZX}=5\mathrm{WU}=5×4=20$

## How scale factor affects dilation

When we dilate an object, we enlarge it. We do this by changing the area with a scale factor of ${k}_{2}$ .

If we were working with a three-dimensional object (such as a model airplane), the scale factor would be ${k}_{3}$ .

## When do we use scale factor?

We can use scale factor to help with a number of real-world problems:

• Creating Models: We can use scale factor to create accurate models. For example, we might want to plan out a new urban center. To make sure everything fits, we would need to create a model with the correct proportions and dimensions in relation to nearby landmarks. We could also make a scale model of something that is far too small to view easily, such as a microscopic organism or a microchip.
• Creating Maps: Scale factor is very useful when creating maps. Most maps have a scale that shows you a representation of a single kilometer scaled down, allowing you to easily visualize real distances.
• Recipes: Scale factor helps us multiply recipes. For example, if a recipe feeds four people and we have 8 guests, we can multiply all of the ingredients by a scale factor of 2.

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