Does there exist an equilateral triangle with sides 50, 50, 50? In general, does there exist an equilateral triangle of any sidelength? Justify your answer.
Yes, an equilateral triangle with sides 50, 50, 50 exists. In general, an equilateral triangle exists for any positive sidelength.
Justification: For an equilateral triangle with side length 's', we check the triangle inequality: The longest side is 's', and the sum of the other two sides is s + s = 2s. Since s < 2s for any positive value of s, the triangle inequality is always satisfied.
For sides 50, 50, 50: Check: Is 50 < 50 + 50? 50 < 100 Yes, the inequality is satisfied.
For general sidelength s: Check: Is s < s + s? s < 2s for all positive s Yes, the inequality is always satisfied.
∴ An equilateral triangle can be constructed with any positive sidelength.
Explanation
The chapter establishes that if a set of three lengths satisfies the triangle inequality, then a triangle exists with those sidelengths. For an equilateral triangle with all sides equal to 's', we need to verify that each side is less than the sum of the other two. Since s < s + s = 2s for any positive s, the triangle inequality is always satisfied. The chapter also shows the construction method using a compass to construct equilateral triangles.
Solution Steps
For equilateral triangle with sides 50, 50, 50:
Check: Is 50 < 50 + 50? Yes, 50 < 100
For general sidelength s:
Check: Is s < s + s? Yes, s < 2s for all positive s
Therefore, equilateral triangle exists for any positive sidelength