An isosceles triangle has two sides of equal length, [tex]$a$[/tex], and a base, [tex]$b$[/tex]. The perimeter of the triangle is 15.7 inches, so the equation to solve is [tex]$2 a+b=15.7$[/tex].
If we recall that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side, which lengths make sense for possible values of [tex]$b$[/tex]? Select two options.
-2 in.
0 in.
0.5 in.
2 in.
7.9 in.
Answer (1)
Express a in terms of b using the perimeter equation: a = 2 15.7 − b .
Apply the triangle inequality: b"> a + a > b , which simplifies to b < 7.85 .
Ensure that 0"> b > 0 and 0"> a > 0 , leading to the condition b < 15.7 .
Check which of the given values of b satisfy the inequality 0 < b < 7.85 , resulting in the valid options: 0.5 , 2 .
Explanation
Problem Analysis Let's analyze the problem. We are given an isosceles triangle with two sides of equal length a and a base b . The perimeter is 15.7 inches, so we have the equation 2 a + b = 15.7 . We need to find which of the given values for b make sense, considering the triangle inequality.
Triangle Inequality The triangle inequality states that the sum of any two sides of a triangle must be greater than the third side. In our case, this means:
b"> a + a > b
a"> a + b > a (which simplifies to 0"> b > 0 )
Also, since a represents a side length, 0"> a > 0 .
Expressing a in terms of b From the perimeter equation, we can express a in terms of b :
2 a + b = 15.7
2 a = 15.7 − b
a = 2 15.7 − b
Substituting into the Inequality Now, substitute this expression for a into the triangle inequality b"> a + a > b :
b"> 2 15.7 − b + 2 15.7 − b > b
b"> 15.7 − b > b
2b"> 15.7 > 2 b
b < 2 15.7
b < 7.85
Additional Conditions We also have the condition that 0"> b > 0 and 0"> a > 0 . Since a = 2 15.7 − b , for a to be greater than 0, we need 0"> 15.7 − b > 0 , which means b < 15.7 . This condition is already satisfied by b < 7.85 .
Checking the Options So, we need to find the values of b from the given options that satisfy 0 < b < 7.85 :
-2 in: Invalid, since b must be greater than 0. 0 in: Invalid, since b must be greater than 0. 0.5 in: Valid, since 0 < 0.5 < 7.85 .
2 in: Valid, since 0 < 2 < 7.85 .
7.9 in: Invalid, since 7.85"> 7.9 > 7.85 .
Final Answer Therefore, the possible values for b are 0.5 inches and 2 inches.
Examples
In architecture, understanding triangle inequalities is crucial when designing triangular structures like trusses. If you're designing a truss with a fixed perimeter, knowing the possible range of the base length ensures the structural integrity of the truss. For instance, if you want a truss with a perimeter of 15.7 meters, you can use the triangle inequality to determine the valid lengths for the base, ensuring the truss can bear the intended load without collapsing. This principle helps architects and engineers create stable and efficient designs.
