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Sagot :
To solve this problem, let's follow these steps:
1. Understanding the given information:
- The triangle is isosceles, meaning two sides are of equal length [tex]\( a \)[/tex] and the base is [tex]\( b \)[/tex].
- The perimeter of the triangle is given as 15.7 inches.
2. Formulating the equation:
- We know that the perimeter of the triangle (the sum of all three sides) is 15.7. Hence, the equation becomes:
[tex]\[ 2a + b = 15.7 \][/tex]
3. Solving for [tex]\( a \)[/tex] in terms of [tex]\( b \)[/tex]:
- Rearrange the equation to express [tex]\( a \)[/tex] in terms of [tex]\( b \)[/tex]:
[tex]\[ 2a = 15.7 - b \][/tex]
[tex]\[ a = \frac{{15.7 - b}}{2} \][/tex]
4. Checking possible values of [tex]\( b \)[/tex]:
- We need to ensure that any selected values of [tex]\( b \)[/tex] satisfy 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.
- Let's evaluate the given options one by one:
Option 1: [tex]\( b = -2 \)[/tex] inches
[tex]\[ a = \frac{{15.7 - (-2)}}{2} = \frac{{15.7 + 2}}{2} = \frac{17.7}{2} = 8.85 \text{ inches} \][/tex]
- For [tex]\( b = -2 \)[/tex], [tex]\( a \)[/tex] is calculated to be 8.85 inches.
- However, since [tex]\( b \)[/tex] cannot be negative in geometry (a side length has to be a positive real number), we discard this option.
Option 2: [tex]\( b = 0 \)[/tex] inches
[tex]\[ a = \frac{{15.7 - 0}}{2} = \frac{15.7}{2} = 7.85 \text{ inches} \][/tex]
- For [tex]\( b = 0 \)[/tex], [tex]\( a \)[/tex] is calculated to be 7.85 inches.
- This means if both equal sides [tex]\( a \)[/tex] are 7.85 inches and the base [tex]\( b \)[/tex] is 0 inches, it would not form a triangle (the base cannot be 0). Hence, we discard this option too.
Option 3: [tex]\( b = 0.5 \)[/tex] inches
[tex]\[ a = \frac{{15.7 - 0.5}}{2} = \frac{15.2}{2} = 7.6 \text{ inches} \][/tex]
- For [tex]\( b = 0.5 \)[/tex], [tex]\( a \)[/tex] is calculated to be 7.6 inches.
- To satisfy the triangle inequality theorem:
[tex]\[ a + a > b \implies 7.6 + 7.6 > 0.5 \implies 15.2 > 0.5 \text{ (true)} \][/tex]
[tex]\[ a + b > a \implies 7.6 + 0.5 > 7.6 \implies 8.1 > 7.6 \text{ (true)} \][/tex]
- These conditions uphold, hence [tex]\( b = 0.5 \)[/tex] is a valid option.
Option 4: [tex]\( b = 2 \)[/tex] inches
[tex]\[ a = \frac{{15.7 - 2}}{2} = \frac{13.7}{2} = 6.85 \text{ inches} \][/tex]
- For [tex]\( b = 2 \)[/tex], [tex]\( a \)[/tex] is calculated to be 6.85 inches.
- To satisfy the triangle inequality theorem:
[tex]\[ a + a > b \implies 6.85 + 6.85 > 2 \implies 13.7 > 2 \text{ (true)} \][/tex]
[tex]\[ a + b > a \implies 6.85 + 2 > 6.85 \implies 8.85 > 6.85 \text{ (true)} \][/tex]
- These conditions uphold, hence [tex]\( b = 2 \)[/tex] is a valid option.
Option 5: [tex]\( b = 7.9 \)[/tex] inches
[tex]\[ a = \frac{{15.7 - 7.9}}{2} = \frac{7.8}{2} = 3.9 \text{ inches} \][/tex]
- For [tex]\( b = 7.9 \)[/tex], [tex]\( a \)[/tex] is calculated to be 3.9 inches.
- To satisfy the triangle inequality theorem:
[tex]\[ a + a > b \implies 3.9 + 3.9 > 7.9 \implies 7.8 > 7.9 \text{ (false)} \][/tex]
- This condition does not hold up (false), hence [tex]\( b = 7.9 \)[/tex] is not a valid option.
5. Conclusion on valid options:
Given the above evaluations, the lengths that make sense for possible values of [tex]\( b \)[/tex] are:
- 0.5 inches
- 2 inches
1. Understanding the given information:
- The triangle is isosceles, meaning two sides are of equal length [tex]\( a \)[/tex] and the base is [tex]\( b \)[/tex].
- The perimeter of the triangle is given as 15.7 inches.
2. Formulating the equation:
- We know that the perimeter of the triangle (the sum of all three sides) is 15.7. Hence, the equation becomes:
[tex]\[ 2a + b = 15.7 \][/tex]
3. Solving for [tex]\( a \)[/tex] in terms of [tex]\( b \)[/tex]:
- Rearrange the equation to express [tex]\( a \)[/tex] in terms of [tex]\( b \)[/tex]:
[tex]\[ 2a = 15.7 - b \][/tex]
[tex]\[ a = \frac{{15.7 - b}}{2} \][/tex]
4. Checking possible values of [tex]\( b \)[/tex]:
- We need to ensure that any selected values of [tex]\( b \)[/tex] satisfy 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.
- Let's evaluate the given options one by one:
Option 1: [tex]\( b = -2 \)[/tex] inches
[tex]\[ a = \frac{{15.7 - (-2)}}{2} = \frac{{15.7 + 2}}{2} = \frac{17.7}{2} = 8.85 \text{ inches} \][/tex]
- For [tex]\( b = -2 \)[/tex], [tex]\( a \)[/tex] is calculated to be 8.85 inches.
- However, since [tex]\( b \)[/tex] cannot be negative in geometry (a side length has to be a positive real number), we discard this option.
Option 2: [tex]\( b = 0 \)[/tex] inches
[tex]\[ a = \frac{{15.7 - 0}}{2} = \frac{15.7}{2} = 7.85 \text{ inches} \][/tex]
- For [tex]\( b = 0 \)[/tex], [tex]\( a \)[/tex] is calculated to be 7.85 inches.
- This means if both equal sides [tex]\( a \)[/tex] are 7.85 inches and the base [tex]\( b \)[/tex] is 0 inches, it would not form a triangle (the base cannot be 0). Hence, we discard this option too.
Option 3: [tex]\( b = 0.5 \)[/tex] inches
[tex]\[ a = \frac{{15.7 - 0.5}}{2} = \frac{15.2}{2} = 7.6 \text{ inches} \][/tex]
- For [tex]\( b = 0.5 \)[/tex], [tex]\( a \)[/tex] is calculated to be 7.6 inches.
- To satisfy the triangle inequality theorem:
[tex]\[ a + a > b \implies 7.6 + 7.6 > 0.5 \implies 15.2 > 0.5 \text{ (true)} \][/tex]
[tex]\[ a + b > a \implies 7.6 + 0.5 > 7.6 \implies 8.1 > 7.6 \text{ (true)} \][/tex]
- These conditions uphold, hence [tex]\( b = 0.5 \)[/tex] is a valid option.
Option 4: [tex]\( b = 2 \)[/tex] inches
[tex]\[ a = \frac{{15.7 - 2}}{2} = \frac{13.7}{2} = 6.85 \text{ inches} \][/tex]
- For [tex]\( b = 2 \)[/tex], [tex]\( a \)[/tex] is calculated to be 6.85 inches.
- To satisfy the triangle inequality theorem:
[tex]\[ a + a > b \implies 6.85 + 6.85 > 2 \implies 13.7 > 2 \text{ (true)} \][/tex]
[tex]\[ a + b > a \implies 6.85 + 2 > 6.85 \implies 8.85 > 6.85 \text{ (true)} \][/tex]
- These conditions uphold, hence [tex]\( b = 2 \)[/tex] is a valid option.
Option 5: [tex]\( b = 7.9 \)[/tex] inches
[tex]\[ a = \frac{{15.7 - 7.9}}{2} = \frac{7.8}{2} = 3.9 \text{ inches} \][/tex]
- For [tex]\( b = 7.9 \)[/tex], [tex]\( a \)[/tex] is calculated to be 3.9 inches.
- To satisfy the triangle inequality theorem:
[tex]\[ a + a > b \implies 3.9 + 3.9 > 7.9 \implies 7.8 > 7.9 \text{ (false)} \][/tex]
- This condition does not hold up (false), hence [tex]\( b = 7.9 \)[/tex] is not a valid option.
5. Conclusion on valid options:
Given the above evaluations, the lengths that make sense for possible values of [tex]\( b \)[/tex] are:
- 0.5 inches
- 2 inches
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