Question
This isosceles triangle has two sides of equal length, \( a \), that are longer than the length of the base, \( b \). The perimeter of the triangle is 15.7 centimeters. The equation \( 2 a+b=15.7 \) can be used to find the side lengths. If one of the longer sides is 6.3 centimeters, what is the length of the base?
Ask by Todd Sandoval. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The base is 3.1 centimeters long.
Solution
To find the length of the base \( b \) of the isosceles triangle, follow these steps:
1. **Understand the Given Information:**
- The triangle has two equal sides, each of length \( a \).
- The base of the triangle has length \( b \).
- The perimeter of the triangle is \( 15.7 \) centimeters.
- One of the equal sides (\( a \)) is \( 6.3 \) centimeters.
2. **Set Up the Perimeter Equation:**
The perimeter \( P \) of the triangle is the sum of all its sides:
\[
P = a + a + b = 2a + b
\]
Given that \( P = 15.7 \) centimeters, the equation becomes:
\[
2a + b = 15.7
\]
3. **Substitute the Known Value of \( a \):**
Substitute \( a = 6.3 \) centimeters into the equation:
\[
2(6.3) + b = 15.7
\]
\[
12.6 + b = 15.7
\]
4. **Solve for \( b \):**
Subtract \( 12.6 \) from both sides of the equation to isolate \( b \):
\[
b = 15.7 - 12.6
\]
\[
b = 3.1 \text{ centimeters}
\]
**Answer:**
The length of the base is \( 3.1 \) centimeters.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Bonus Knowledge
To find the length of the base \( b \) when one of the longer sides \( a \) is 6.3 centimeters, we can substitute \( a \) into the perimeter equation \( 2a + b = 15.7 \). Substituting \( a = 6.3 \): \[ 2(6.3) + b = 15.7 \] Calculating \( 2(6.3) \): \[ 12.6 + b = 15.7 \] Now, solving for \( b \): \[ b = 15.7 - 12.6 = 3.1 \] So, the length of the base \( b \) is 3.1 centimeters!
