3. Trigonometry

d. Trig Identities

6. The Triangle Inequality

Given a triangle with side lengths \(a\), \(b\) and \(c\), then: \[\begin{aligned} |a-b| \le c \le a+b \end{aligned}\] \(\Leftarrow\Leftarrow\) Read it! It's animated.

Consider a triangle with sides of lengths \(a\), \(b\) and \(c\). We will hold the lengths of \(a\) and \(b\) fixed while the angle between them changes. Then the length \(c\) changes.

pf_TriIneq1

When the angle between sides \(a\) and \(b\) is very small, side \(c\) approaches \(|a-b|\) but always stays larger than this as long as the angle is non-zero. So: \[ c \ge |a-b| \]

pf_TriIneq2

Because we don't know which is bigger, \(a\) or \(b\), and \(c\) must be bigger than the positive difference. The figure above has \(b \lt a\). The figure at the right has \(b \gt a\).

pf_TriIneq2b

When the angle between sides \(a\) and \(b\) is very large, side \(c\) approaches \(a+b\) but always stays less than this as long as the angle is not \(180^\circ\). So: \[ c \le a+b \]

pf_TriIneq3

Together we have \[ |a-b| \le c \le a+b \]

Given three positive numbers \(a\), \(b\) and \(c\). If \(c\) is the largest and \(c \lt a+b\), then there exists a triangle with sides \(a\), \(b\) and \(c\).
\(\Leftarrow\Leftarrow\) Read it! It's easy.

Draw a line segment \(\overline{AB}\) of length \(c\).
At the end \(A\), draw a circle of radius \(b\).
At the end \(B\), draw a circle of radius \(a\).
Since \(a+b \gt c\), these circles must intersect at two points \(C\) and \(D\). Then the triangle \(\triangle ABC\) has sides \(a\), \(b\) and \(c\).

pf_TriExist

In each problem, is there a triangle with sides \(a\), \(b\) and \(c\)?

  1. \(\quad a=3 \qquad b=4 \qquad c=6\)

    Correct! Since \(6\) is the longest side and \(6 \lt 3+4\), there exists a triangle with sides \(3\), \(4\) and \(6\).

    Incorrect! Since \(6\) is the longest side and \(6 \lt 3+4\), there does exists a triangle with sides \(3\), \(4\) and \(6\).

  2. \(\quad a=3 \qquad b=4 \qquad c=8\)

    Incorrect! Since \(8 \gt 3+4\), a triangle with sides \(3\), \(4\) and \(8\) would violate the Triangle Inequality.

    Correct! Since \(8 \gt 3+4\), a triangle with sides \(3\), \(4\) and \(8\) would violate the Triangle Inequality.

  3. \(\quad a=9 \qquad b=2 \qquad c=6\)

    Incorrect! Since \(9 \gt 2+6\), a triangle with sides \(9\), \(2\) and \(6\) would violate the Triangle Inequality.

    Correct! Since \(9 \gt 2+6\), a triangle with sides \(9\), \(2\) and \(6\) would violate the Triangle Inequality.

  4. \(\quad a=7 \qquad b=2 \qquad c=6\)

    Correct! Since \(7\) is the longest side and \(7 \lt 2+6\), there exists a triangle with sides \(7\), \(2\) and \(6\).

    Incorrect! Since \(7\) is the longest side and \(7 \lt 2+6\), there does exists a triangle with sides \(7\), \(2\) and \(6\).

  5. \(\quad a=4 \qquad b=10 \qquad c=7\)

    Correct! Since \(10\) is the longest side and \(10 \lt 4+7\), there exists a triangle with sides \(4\), \(10\) and \(7\).

    Incorrect! Since \(10\) is the longest side and \(10 \lt 4+7\), there does exists a triangle with sides \(4\), \(10\) and \(7\).

  6. \(\quad a=4 \qquad b=12 \qquad c=7\)

    Incorrect! Since \(12 \gt 4+7\), a triangle with sides \(4\), \(12\) and \(7\) would violate the Triangle Inequality.

    Correct! Since \(12 \gt 4+7\), a triangle with sides \(4\), \(12\) and \(7\) would violate the Triangle Inequality.

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