To find the measures of each side of the triangle given the ratio of the sides \(7:9:12\) and a perimeter of \(84\) inches, we can follow these steps:
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Identify the ratio parts: Let the sides of the triangle be represented as \(7x\), \(9x\), and \(12x\), where \(x\) is a common multiplier.
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Set up the equation using the perimeter: The perimeter of the triangle is the sum of the lengths of its sides: \[ 7x + 9x + 12x = 84 \]
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Combine the terms: Combine the terms on the left side: \[ (7 + 9 + 12)x = 84 \] This simplifies to: \[ 28x = 84 \]
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Solve for \(x\): Divide both sides by the coefficient of \(x\): \[ x = \frac{84}{28} = 3 \]
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Calculate the lengths of each side: Now substitute \(x\) back into the expressions for each side:
- First side: \[ 7x = 7 \times 3 = 21 \text{ inches} \]
- Second side: \[ 9x = 9 \times 3 = 27 \text{ inches} \]
- Third side: \[ 12x = 12 \times 3 = 36 \text{ inches} \]
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Present the measures of each side: The measures of the sides of the triangle are:
- First side: \(21\) inches
- Second side: \(27\) inches
- Third side: \(36\) inches
Thus, the sides of the triangle are 21 inches, 27 inches, and 36 inches.