The ratio of the measures of the sides of a triangle is 3:4:6, and its perimeter is 104 feet. Find the measure of each side of the triangle.

1 answer

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:

  1. First side: \[ 3x = 3 \times 8 = 24 , \text{feet} \]

  2. Second side: \[ 4x = 4 \times 8 = 32 , \text{feet} \]

  3. 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:

\[ \boxed{24 , \text{feet}, 32 , \text{feet}, 48 , \text{feet}} \]