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## Geometric Inequalities

Given 5 sticks of length 1, 3, 5, 9, and 10, how many distinct triangles can be formed? Learn the techniques and develop an intuition for working with geometric inequalities.

# Triangle Inequality

The side lengths of a non-degenerate $$\triangle ABC$$ are $$4, 11$$ and $$c.$$ Which of the following is a possible value for $$c?$$

Two legs of a triangle have lengths of 7.4 and 17.3 respectively. Given that the length of the third side is a whole number, what is the largest possible length for the third side?

The three side lengths of a non-degenerate triangle are $$x, x + 5$$ and $$x + 11.$$ Which of the following is NOT a possible value for $$x?$$

Three positive integers $$a, b$$ and $$c$$ are the side lengths of $$\triangle ABC.$$ If $$c$$ is the longest side and $$a + b + c = 21,$$ what is the maximum value of $$c?$$

Two of the legs of an isosceles triangle have length 12 and 27 respectively. What is the perimeter of the triangle?

Details and assumptions

An isosceles triangle is a triangle with exactly two equal sides.

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