Question
Turn this into a short little paragraph:
Detine the variable for the third side length.
Let the length of the third side be denoted as x
2
Apply the triangle inequality theorem.
This theorem states that the suraof the lengths of any two sides of a triangle must be greater than the length of the remaining side. Therefore, the following inequalities must hold:
8 + 12> x = 20 > x
8+x>12 = x > 4
12+x > 8 → x>-4
(which is always true since side lengths are positive
3 Combine the inequalities to find the range of possible x
From the above, the combined condition is:
4 < X < 20
4 Evaluate the given options against the inequality range.
Options: 20 cm, 24 cm, 3 cm, 5 cm.
Only the value 5 cm satisfies the condition 4 < x < 20
Detine the variable for the third side length.
Let the length of the third side be denoted as x
2
Apply the triangle inequality theorem.
This theorem states that the suraof the lengths of any two sides of a triangle must be greater than the length of the remaining side. Therefore, the following inequalities must hold:
8 + 12> x = 20 > x
8+x>12 = x > 4
12+x > 8 → x>-4
(which is always true since side lengths are positive
3 Combine the inequalities to find the range of possible x
From the above, the combined condition is:
4 < X < 20
4 Evaluate the given options against the inequality range.
Options: 20 cm, 24 cm, 3 cm, 5 cm.
Only the value 5 cm satisfies the condition 4 < x < 20
Answers
Answered by
GPT-4o mini
To determine the length of the third side of a triangle, we denote it as \( x \) and apply the triangle inequality theorem, which states that the sum of the lengths of any two sides must be greater than the length of the remaining side. This results in the following inequalities: \( 8 + 12 > x \) (which simplifies to \( x < 20 \)), \( 8 + x > 12 \) (leading to \( x > 4 \)), and \( 12 + x > 8 \) (which is always true since side lengths are positive). Combining these inequalities gives the range \( 4 < x < 20 \). Evaluating the options—20 cm, 24 cm, 3 cm, and 5 cm—only the value 5 cm satisfies the condition, as it falls within the established range.