Westonci.ca offers quick and accurate answers to your questions. Join our community and get the insights you need today. Discover reliable solutions to your questions from a wide network of experts on our comprehensive Q&A platform. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
To determine the range of possible values for the third side of the triangle, we need to use the Triangle Inequality Theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let's denote the sides of the triangle as [tex]\(a = 2\)[/tex], [tex]\(b = 7\)[/tex], and [tex]\(x\)[/tex] as the unknown length of the third side. The Triangle Inequality Theorem gives us three inequalities:
1. [tex]\(a + b > x\)[/tex]
2. [tex]\(a + x > b\)[/tex]
3. [tex]\(b + x > a\)[/tex]
Substituting the known values into these inequalities, we get:
1. [tex]\(2 + 7 > x\)[/tex], which simplifies to [tex]\(9 > x\)[/tex] or [tex]\(x < 9\)[/tex].
2. [tex]\(2 + x > 7\)[/tex], which simplifies to [tex]\(x > 5\)[/tex].
3. [tex]\(7 + x > 2\)[/tex], which simplifies to [tex]\(7 + x > 2\)[/tex], which is always true for any positive value of [tex]\(x\)[/tex] and does not provide a new constraint.
Combining these inequalities, we have:
[tex]\[5 < x < 9\][/tex]
So the range of possible values for [tex]\(x\)[/tex], the third side of the triangle, is given by the inequality:
[tex]\[5 < x < 9\][/tex]
Thus, the correct answer is:
C. [tex]\(5 < x < 9\)[/tex]
Let's denote the sides of the triangle as [tex]\(a = 2\)[/tex], [tex]\(b = 7\)[/tex], and [tex]\(x\)[/tex] as the unknown length of the third side. The Triangle Inequality Theorem gives us three inequalities:
1. [tex]\(a + b > x\)[/tex]
2. [tex]\(a + x > b\)[/tex]
3. [tex]\(b + x > a\)[/tex]
Substituting the known values into these inequalities, we get:
1. [tex]\(2 + 7 > x\)[/tex], which simplifies to [tex]\(9 > x\)[/tex] or [tex]\(x < 9\)[/tex].
2. [tex]\(2 + x > 7\)[/tex], which simplifies to [tex]\(x > 5\)[/tex].
3. [tex]\(7 + x > 2\)[/tex], which simplifies to [tex]\(7 + x > 2\)[/tex], which is always true for any positive value of [tex]\(x\)[/tex] and does not provide a new constraint.
Combining these inequalities, we have:
[tex]\[5 < x < 9\][/tex]
So the range of possible values for [tex]\(x\)[/tex], the third side of the triangle, is given by the inequality:
[tex]\[5 < x < 9\][/tex]
Thus, the correct answer is:
C. [tex]\(5 < x < 9\)[/tex]
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.