In an equilateral triangle, all three sides are equal. Given that the sides of the triangle are \(4a\), \(3b\), and \( (a + b + c) \), we can set up the following equations based on the property of equilateral triangles:
- \( 4a = 3b \)
- \( 4a = a + b + c \)
- \( 3b = a + b + c \)
Now, let's solve these equations step by step.
From Equation 1:
From \( 4a = 3b \), we can express \( b \) in terms of \( a \):
\[ b = \frac{4a}{3} \]
Substitute \( b \) into Equations 2 and 3:
Substituting into Equation 2:
Now, substitute \( b = \frac{4a}{3} \) into \( 4a = a + b + c \):
\[ 4a = a + \frac{4a}{3} + c \]
To combine terms, multiply through by 3 to eliminate the fraction:
\[ 12a = 3a + 4a + 3c \]
Simplifying gives:
\[ 12a = 7a + 3c \]
Rearranging gives:
\[ 5a = 3c \quad \Rightarrow \quad c = \frac{5a}{3} \]
Substituting into Equation 3:
Now substitute \( b = \frac{4a}{3} \) and \( c = \frac{5a}{3} \) into \( 3b = a + b + c \):
\[ 3\left(\frac{4a}{3}\right) = a + \frac{4a}{3} + \frac{5a}{3} \]
This simplifies to:
\[ 4a = a + \frac{9a}{3} \]
Converting \( \frac{9a}{3} \) to a simpler form gives:
\[ 4a = a + 3a \]
Indeed, this confirms:
\[ 4a = 4a \]
Now, we can find the triangle side lengths
We know:
- \( b = \frac{4a}{3} \)
- \( c = \frac{5a}{3} \)
Let's find the lengths:
To find the lengths, let's test the value of \( a \) and find \( b \):
Let \( a = 3 \), then:
\[ b = \frac{4 \times 3}{3} = 4 \]
So, the lengths of the triangle sides would be:
- \( 4a = 4 \times 3 = 12 \) cm
- \( 3b = 3 \times 4 = 12 \) cm
- \( a + b + c = 3 + 4 + \frac{5 \times 3}{3} = 3 + 4 + 5 = 12 \) cm
Thus, all side lengths of the equilateral triangle will confirm out to be 12 cm each:
Conclusion
The lengths of the triangle sides are:
\[ \text{Each side length} = 12 \text{ cm} \]