To find the measures of the sides of the triangle, we first denote the lengths of the sides in accordance with the given ratio 3:4:6. Let:
- \( 3x \) be the length of the first side,
- \( 4x \) be the length of the second side,
- \( 6x \) be the length of the third side.
The perimeter of the triangle is the sum of all its sides. Thus, we have:
\[ 3x + 4x + 6x = 104 , \text{feet} \]
Combining the terms on the left side:
\[ 13x = 104 \]
Next, we solve for \( x \):
\[ x = \frac{104}{13} = 8 \]
Now that we have the value of \( x \), we can find the lengths of each side:
-
First side: \[ 3x = 3 \times 8 = 24 , \text{feet} \]
-
Second side: \[ 4x = 4 \times 8 = 32 , \text{feet} \]
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Third side: \[ 6x = 6 \times 8 = 48 , \text{feet} \]
Thus, the lengths of the sides of the triangle are:
- First side: 24 feet
- Second side: 32 feet
- Third side: 48 feet
To confirm, we check the perimeter:
\[ 24 + 32 + 48 = 104 , \text{feet} \]
Everything checks out, so the measures of the sides of the triangle are:
\[ \boxed{24 , \text{feet}, 32 , \text{feet}, 48 , \text{feet}} \]
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