Triangle Inequality Theorem: To determine if lengths can form a triangle, use the Triangle Inequality Theorem which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
First Set of Lengths: Check the first set of lengths: $6$ , $3$ , $21$ . $\newline$ Calculate if $6 + 3 > 21$ .
Second Set of Lengths: $6 + 3 = 9$ , which is not greater than $21$ . $\newline$ So, the lengths $6$ , $3$ , and $21$ cannot form a triangle.
Third Set of Lengths: Check the second set of lengths: $21$ , $11$ , $3$ . $\newline$ Calculate if $21 + 11 > 3$ .
Fourth Set of Lengths: $21 + 11 = 32$ , which is greater than $3$ . Now check the other combinations: $21 + 3 > 11$ and $11 + 3 > 21$ .
Fourth Set of Lengths: $21 + 11 = 32$ , which is greater than $3$ . Now check the other combinations: $21 + 3 > 11$ and $11 + 3 > 21$ . $21 + 3 = 24$ , which is greater than $11$ . $11 + 3 = 14$ , which is not greater than $21$ . So, the lengths $21$ , $11$ , and $3$ cannot form a triangle.
Fourth Set of Lengths: $21 + 11 = 32$ , which is greater than $3$ . Now check the other combinations: $21 + 3 > 11$ and $11 + 3 > 21$ . $21 + 3 = 24$ , which is greater than $11$ . $11 + 3 = 14$ , which is not greater than $21$ . So, the lengths $21$ , $11$ , and $3$ cannot form a triangle. Check the third set of lengths: $3$ $1$ , $3$ $2$ , $3$ $3$ . Calculate if $3$ $4$ .
Fourth Set of Lengths: $21 + 11 = 32$ , which is greater than $3$ . Now check the other combinations: $21 + 3 > 11$ and $11 + 3 > 21$ . $21 + 3 = 24$ , which is greater than $11$ . $11 + 3 = 14$ , which is not greater than $21$ . So, the lengths $21$ , $11$ , and $3$ cannot form a triangle. Check the third set of lengths: $3$ $1$ , $3$ $2$ , $3$ $3$ . Calculate if $3$ $4$ . $3$ $5$ , which is greater than $3$ $3$ . Now check the other combinations: $3$ $7$ and $3$ $8$ .
Fourth Set of Lengths: $21 + 11 = 32$ , which is greater than $3$ . Now check the other combinations: $21 + 3 > 11$ and $11 + 3 > 21$ . $21 + 3 = 24$ , which is greater than $11$ . $11 + 3 = 14$ , which is not greater than $21$ . So, the lengths $21$ , $11$ , and $3$ cannot form a triangle. Check the third set of lengths: $3$ $1$ , $3$ $2$ , $3$ $3$ . Calculate if $3$ $4$ . $3$ $5$ , which is greater than $3$ $3$ . Now check the other combinations: $3$ $7$ and $3$ $8$ . $3$ $9$ , which is not greater than $3$ $2$ . So, the lengths $3$ $1$ , $3$ $2$ , and $3$ $3$ cannot form a triangle.
Fourth Set of Lengths: $21 + 11 = 32$ , which is greater than $3$ . Now check the other combinations: $21 + 3 > 11$ and $11 + 3 > 21$ . $21 + 3 = 24$ , which is greater than $11$ . $11 + 3 = 14$ , which is not greater than $21$ . So, the lengths $21, 11,$ and $3$ cannot form a triangle. Check the third set of lengths: $3$ $0$ . Calculate if $3$ $1$ . $3$ $2$ , which is greater than $3$ $3$ . Now check the other combinations: $3$ $4$ and $3$ $5$ . $3$ $6$ , which is not greater than $3$ $7$ . So, the lengths $3$ $8$ and $3$ $3$ cannot form a triangle. Check the fourth set of lengths: $21 + 3 > 11$ $0$ . Calculate if $21 + 3 > 11$ $1$ .
Fourth Set of Lengths: $21 + 11 = 32$ , which is greater than $3$ . Now check the other combinations: $21 + 3 > 11$ and $11 + 3 > 21$ . $21 + 3 = 24$ , which is greater than $11$ . $11 + 3 = 14$ , which is not greater than $21$ . So, the lengths $21$ , $11$ , and $3$ cannot form a triangle. Check the third set of lengths: $3$ $1$ , $3$ $2$ , $3$ $3$ . Calculate if $3$ $4$ . $3$ $5$ , which is greater than $3$ $3$ . Now check the other combinations: $3$ $7$ and $3$ $8$ . $3$ $9$ , which is not greater than $3$ $2$ . So, the lengths $3$ $1$ , $3$ $2$ , and $3$ $3$ cannot form a triangle. Check the fourth set of lengths: $21 + 3 > 11$ $4$ , $21 + 3 > 11$ $5$ , $21 + 3 > 11$ $6$ . Calculate if $21 + 3 > 11$ $7$ . $21 + 3 > 11$ $8$ , which is not greater than $21 + 3 > 11$ $6$ . So, the lengths $21 + 3 > 11$ $4$ , $21 + 3 > 11$ $5$ , and $21 + 3 > 11$ $6$ cannot form a triangle.