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Triangle inequality
The triangle inequality is a fundamental geometric property used to determine whether a triangle can exist given three specific side lengths.
A set of three lengths satisfies the triangle inequality if each length is smaller than the sum of the other two lengths.
If a set of three lengths satisfies this condition, a triangle can be constructed; if not, construction is impossible.
Here \(AB = c\), \(BC = a\) and \(CA = b\).
Thus, the above inequality can be written in the notation as follows:
\(a + b > c\)
\(b + c > a\) and
\(c + a > b\).
Reasoning using "Direct path vs Roundabout path"
For a triangle with vertices \(A\), \(B\), \(C\):
- Direct path from \(B\) to \(C =\) length \(BC\)
- Roundabout path from \(B\) to \(C\) via \(A = BA + AC\)
For the triangle to be possible, the direct path between any two vertices must be shorter than the roundabout path via the third vertex. This must hold for all three pairs of vertices simultaneously.
Example 1: (triangle exists): Lengths \(6 \ cm\), \(8 \ cm\), \(10 \ cm\)
- \(BC = 6\) vs \(BA + AC = 8 + 10 = 18\) → direct path is shorter \(✔\)
- \(AB = 8\) vs \(AC + CB = 10 + 6 = 16\) → direct path is shorter \(✔\)
- \(CA = 10\) vs \(CB + BA = 6 + 8 = 14\) → direct path is shorter \(✔\)
Since the sidelengths satisfy the triangle inequality criteria, \(ABC\) forms a triangle.
Example 2: (triangle does NOT exist): Lengths \(10 \ cm\), \(15 \ cm\), \(30 \ cm\)
- \(BC = 10\) vs \(BA + AC = 15 + 30 = 45\) → direct path is shorter \(✔\)
- \(AB = 15\) vs \(AC + CB = 30 + 10 = 40\) → direct path is shorter \(✔\)
- \(CA = 30\) vs \(CB + BA = 10 + 15 = 25\) → direct path is longer \(✘\)
Since one comparison fails, such a triangle cannot exist.
How to Test for Existence
Identify the longest length and check if it is less than the sum of the two smaller lengths.
Example:
A set of lengths \(9, 12, 15\) satisfies the triangle inequality because \(15 < 9 + 12\).
Conversely, the set \(10, 15, 30\) does not, because \(30 > 10 + 15\).
Visualising the construction of circles
The Three Possible Cases When Two Circles Are Drawn:
Case 1: Circles touch each other: The sum of the two smaller lengths equals the longest length. The circles meet at a single point on the base, so no triangle is formed.
Case 2: Circles do not intersect: The sum of the two smaller lengths is less than the longest length. The arcs will never meet, making a triangle impossible.
Case 3: Circles intersect internally: The sum of the two smaller lengths is greater than the longest length. The circles intersect at two points, providing the third vertex needed to form a triangle.
Important!
Suppose \(a\), \(b\) and \(c\) are the sides of a triangle with \(a\) and \(b\) are known sides and \(c\) is unknown. Let's use the triangle inequality to find the third side \(c\). As the sum of two sides\((a+b)\) of a triangle is always greater than the third side\((c)\), and difference of two sides\((a−b)\) of a triangle is always lesser than the third side\((c)\), the length of the third side\((c)\) must lie between difference\((a−b)\) and sum\((a+b)\).
That is, \(|a - b| < c < (a + b)\).
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