Exercise 1 Page 453

We were given a triangle $X Y Z .$

Let's suppose that $△ A B C$ is similar to $△ X Y Z .$ Notice that by similarity properties the shortest sides of two similar triangles correspond to each other. Similarly, the longest sides also correspond to each other.

Side of $△ X Y Z$	Corresponds to
Shortest side of $△ X Y Z$ $(10)$	Shortest side of $△ A B C$ $(ℓ_{1})$
Longest side of $△ X Y Z$ $(13)$	Longest side of $△ A B C$ $(ℓ_{2})$
Third side of $△ X Y Z$ $(12)$	Third side of $△ A B C$ $(ℓ_{3})$

We have three ways to find the scale factor.

If we are given the shortest side of $△ A B C,$ the scale factor is $k = \frac{ℓ _{1}}{1 0} .$
If we are given the longest side of $△ A B C,$ the scale factor is $k = \frac{ℓ _{2}}{1 3} .$
If we are given the third side of $△ A B C,$ the scale factor is $k = \frac{ℓ _{3}}{1 2} .$

On the other hand, if we are given one side of

△ A B C

— let's say

A B

— but we are not told which side is

(ℓ_{1},

ℓ_{2},

or

ℓ_{3}),

then we will not know which correspondence to use and we can get an incorrect scale factor.

k_{1} = \frac{A B}{1 0} or k_{2} = \frac{A B}{1 3} or k_{3} = \frac{A B}{1 2}

That is why we needed to know which side is

20

units long — so that we found the correct corresponding scale factor and then the length of the other two sides.