Geometric Constraints Math Example 4

Follow the full solution, then compare it with the other examples linked below.

hard

A triangle has sides

a

b

c

with perimeter

30

. Write all the geometric constraints (triangle inequality and perimeter) that

a

b

c

must satisfy, then find the maximum possible value of side

a

given

b = c

1
Step 1: Perimeter constraint: $a + b + c = 30$ . Triangle inequalities: $a + b > c$ , $a + c > b$ , $b + c > a$ .
2
Step 2: With $b = c$ : $a + 2b = 30 \Rightarrow b = (30-a)/2$ . The binding triangle inequality is $b + c > a \Rightarrow 2b > a \Rightarrow 30 - a > a \Rightarrow a < 15$ .
3
Step 3: Also $a > 0$ . So $0 < a < 15$ . The maximum value $a$ can approach is $15$ (but not reach it — the triangle degenerates).

Answer

a

can approach but never reach

15

cm; the constraint is

0 < a < 15

The triangle inequality is a fundamental geometric constraint. With

b = c

and fixed perimeter, the isoceles condition reduces the problem to a single inequality

a < 15

. As

a \to 15

, the triangle degenerates to a line segment.

About Geometric Constraints

Conditions that limit or restrict the possible positions, sizes, or shapes of geometric objects in a problem.

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