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.

Question

Homi · Accepted Answer

A triangle needs a special rule. The sum of two sides must be greater than the third side.

Step 1 — Checking for sides 50, 50, 50

Let us check a triangle with sides 50, 50, 50. We use the triangle inequality rule. This rule states a condition. The sum of any two sides must be larger than the third side.

Let the sides be $a$ , $b$ , and $c$ . We must check three conditions.

$a + b > c$
$a + c > b$
$b + c > a$

For our triangle, all sides are 50. So, $a = \mathbf{50}$ , $b = \mathbf{50}$ , $c = \mathbf{50}$ .

Let us check the first condition: $a + b > c$ $50 + 50 > 50$ $100 > 50$ This condition is True.

Let us check the second condition: $a + c > b$ $50 + 50 > 50$ $100 > 50$ This condition is True.

Let us check the third condition: $b + c > a$ $50 + 50 > 50$ $100 > 50$ This condition is True.

All three conditions are true. So, such a triangle can exist.

$\boxed{\text{Yes, an equilateral triangle with sides 50, 50, 50 exists.}}$

Diagram 1

Step 2 — Checking for any sidelength

Now, let us consider any equilateral triangle. Let its sidelength be $x$ . All three sides are $x$ , $x$ , $x$ .

We use the triangle inequality rule again. The sum of two sides must be greater than the third side.

Let us check this condition: $x + x > x$ $2x > x$

For this to be true, $x$ must be a positive number. If $x$ is positive, then $2x$ is always greater than $x$ . For example, if $x = 1$ , then $2 > 1$ . This is true. If $x = 0.5$ , then $1 > 0.5$ . This is true. A side length cannot be zero. It cannot be a negative number either. So, for any positive sidelength $x$ , an equilateral triangle exists.

$\boxed{\text{Yes, an equilateral triangle exists for any positive sidelength.}}$

Answer

(i) Yes, an equilateral triangle with sides 50, 50, 50 exists. The sum of any two sides is 100. This is greater than the third side, 50. (ii) Yes, an equilateral triangle always exists for any positive sidelength. Let the sidelength be $x$ . Then $x$ must be a positive number. The condition $x + x > x$ is always true. This means $2x > x$ . This holds for any positive $x$ .

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