# Scale and Indirect Measurement Applications

## Key Questions

• A proportion is a situation when you have two ratios equal to each other. Now, a scale is just a specific type of ratio. When you are given a scale, it will independently make up one part of the proportion (or one side of the equal sign). The other side you will have to set up using the other given information.

Let's say our scale is a ratio of $a$ to $b$. You can also write this as $a : b$ or $\frac{a}{b}$. Now, let's say we are given a certain value $x$ in the same units as $a$, and we are told to find $y$, which is in the same units as $b$. You have to set up a proportion;

When you set up a proportion, you set two ratios equal to each other. But you have to make sure to get it the right way around! Basically, you want to have the same ratio of units on each side of the equation. So to solve our problem with variables, our proportion would look like this:

$\frac{a}{b} = \frac{x}{y}$ But remember, you can always write this as $\frac{b}{a} = \frac{y}{x}$ Both are equally true...you just have to make sure you have consistent units in the numerator row and the denominator row.

In this case, $a , b$ and $x$ are already known. We just have to solve for $y$. We can do this by cross multiplying:

$\frac{a}{b} = \frac{x}{y}$
$a y = b x$
$y = \frac{b x}{a}$

Also, you can always make sure if your proportions are equal in the end by simplifying one of the answers numerically to see if it equates to the other.

Example
On a map, there is a scale of $\frac{1 c m}{5 k m}$. You measure out $3.5 c m$ on the map. How many kilometers in reality is that?

So we already have a scale. It doesn't matter which way we write it, as long as we keep the other side consistent with it. Since $c m$ are our given units, we need to keep that on the same row as the $c m$ in the scale.

(1 cm)/(5km)=(3.5 cm)/(? km)

Cross multiplying, we get:
?km= (3.5cm * (5km)/(1cm))= 17.5 km

This makes sense because our units cancel out- and 17.5/3.5=5 /1 if you divide it numerically.

*note: You need to know three pieces of information to solve proportions like this- but that doesn't always mean knowing the scale itself. If part of the scale was missing, and you knew the other three parts of the proportion, you could still solve.

• You can set up proportions with similar triangles by taking advantage of their side ratios.

By definition, similar triangles have the same angle measures for their corresponding angles, and therefore the corresponding sides have a ratio to them.

For examplle consider the triangles below:

It is given that their corresponding angles have the same measurement, so therefore we can say that they are similar.

Now if we were asked to solve for side ED, then we could do so by setting up a proportion using the side ratios as follows:

$\frac{A C}{D F} = \frac{C B}{F E} = \frac{B A}{D E}$

Now we can just plug in the lengths for the respective sides:

$\frac{7}{14} = \frac{6}{12} = \frac{10}{D E}$

Now we can just simplify and solve:

$\frac{1}{2} = \frac{1}{2} = \frac{10}{D E}$

$\frac{10}{D E} = \frac{1}{2}$

$D E = 2 \left(10\right) = 20$

This is how we can use proportions to solve for side lengths in similar triangles.

Just make sure you set up your proportions with the corresponding sides, or your ratios might come out wrong.

Just as a fun fact, the fact that side lengths have ratios when the angles are the same is fundamental in trigonometry as well, as you use side ratios there too.

Hope that helped :)

A scale is always in the form $1 : n$

A ratio compares two or more quantities.

#### Explanation:

A scale is a specific type of ratio.

It compares a unit on a map or plan or on a scaled model to an actual distance on a map or the size of a real object.

A scale is given in the form: $\text{ "1" ":" } n$

On a map, a scale of $\text{ "1:50" }$ means

$\text{ } 1 c m$ on the map represents $50 c m$ on the ground (in reality)

or $\text{ "200mm" }$ on a plan to this scale, indicates a length of
$200 \times 50 = 10 , 000 m m$ in reality.

A ratio is a comparison between two or more quantities of the same type.

e.g: girls : boys are in the ratio $4 : 3$ means that:

for every $4$ girls there are $3$ boys.

If there are $35$ children in a class with this ratio, we have:

$\frac{4}{7}$ of the class are girls (which means $\frac{4}{7} \times 35 = 20$ girls.)

$\frac{3}{4}$ of the class are boys (which means $\frac{3}{7} \times 35 = 15$ boys.)