Question

1. two sides of a triangle measure 9 cm and 15 cm. which expression represents all possible lengths, x, of the third side? a. x < 6 b. 6 < x < 24 x > 24 d. 9 < x < 15

Explanation:

Step1: Recall triangle inequality rule

The sum of any two sides of a triangle must be greater than the third side. For sides $9$, $15$, and $x$, we get three inequalities:

1. $9 + 15 > x$
2. $9 + x > 15$
3. $15 + x > 9$

Step2: Simplify first inequality

$9 + 15 > x$
$\implies 24 > x$ or $x < 24$

Step3: Simplify second inequality

$9 + x > 15$
$\implies x > 15 - 9$
$\implies x > 6$

Step4: Simplify third inequality

$15 + x > 9$
$\implies x > 9 - 15$
$\implies x > -6$
Since side length is positive, this inequality is irrelevant.

Step5: Combine valid inequalities

Combine $x > 6$ and $x < 24$ to get the range for $x$.

Answer:

B. $6 < x < 24$