Enlargements/reductions

Enlargement and reduction amount to multiplication of a number by a scale factor, which can be any sort of number, even a fraction or decimal number. Enlargements correspond to a scale factor above 1, while reductions correspond to scale factors below 1. Enlargements and reductions are important for the following concepts in this post.

Scale diagrams and scales

A scale diagram is a scale image of an object. Examples of scale diagrams include blueprints, but for the sake of this post, I’ll use an image of a polygon.

This image is a scaled-up version of this object.

The scale factor for the image is 2. We can tell by dividing a side of the image by the corresponding original side. Similarly, if we want to determine the image or original with the scale factor, we can multiply the original by the scale factor or divide the image by the scale factor respectively.

Here, the scale is 2:1, where 2 is the size of the image compared to 1, the size of the original.

Solving proportional equations

Proportional equations are equations with two fractions where one number is missing, and the fractions are equal. Here is an example.

=

Here, represents a side of similar shapes, one of which is missing, and is the ratio.

To solve proportional equations, we can do a cross-multiply, where a denominator is multiplied by the opposite fraction’s numerator and vice versa.

x 4 = 4x

8 6 = 48

4x = 48

From here, we divide by 4 to discover the value of x.

4x 4 = x

48 4 = 12

x = 12

Similar shapes

These are similar shapes.

All of these shapes have the same angles, but different side lengths. This makes them similar shapes.

Similar shapes will always have the same angles and different side lengths. All side lengths can be multiplied to produce the lengths of other shapes that are similar.

Finding missing sides of similar shapes

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Indirect measurement with similar triangles

Thanks to the principles of similar triangles, it is possible to indirectly measure tall objects outside in the sun/using a mirror.

Shadow method

The shadow method requires measuring one’s height and shadow, and the shadow of an object. Divide the length of the object shadow by your shadow, then multiply that by your height.

Mirror method

Using a mirror, set it down on a flat surface, in line with the base of the object that you want to measure. Pick a point on the mirror and move yourself to line up the image of the object with that point on the mirror. Measure the mirror’s distance to you, your eye level height and the distance to the object from the mirror.