Question regarding Triangle Inequality.

Question

Working on the book: Lang, Serge & Murrow, Gene. "Geometry - Second Edition" (p. 23)

If two sides of a triangle are 12 cm and 20 cm, the third side must be larger than __ cm, and smaller than __ cm.

The answer given by the author is:

$20 - 12 < x < 12 + 20$, where x is the length of the third side.

Triangle Inequality. Let $P, Q, M$ be points. Then $d(P, M) < d(P, Q) + d(Q, M)$.

What calculations did the author perform in order to achieve the subtraction on the left side of $20 - 12 < x < 12 + 20$ ?

Answer 1

It's from $20\lt x+12$; subtract $12$ from both sides.

Answer 2

Let the triangle's vertices be $A,B,C$. If $B$ is $12$ cm from $A$, and $C$ is $20$ cm from $B$, then $C$ is $(+12)+(+20) = 12+20$ cm from $A$. This forms the extreme case, which is a degenerate triangle with all sides being collinear.

Now if $B$ is $12$ cm from $A$, but $C$ is $20$ cm from $B$ in the opposite direction to $AB$, then $C$ is $(+12)+(-20) = 12-20$ cm from $A$ (displacement). This is another extreme case where we have another degenerate triangle.

It should not be too hard to imagine that all other triangles fall somewhere in between. Thus $20 - 12 < x < 20 + 12$.