\( x^{2} \) Algebra The perimeter of \( \square P Q R S \) is 84 . Find the length of each side of \( \square P Q R S \) under the given conditions. \( \begin{array}{llll}\text { 27. } P Q=Q R & \text { 28. } Q R=3(R S) & \text { 29. } R S=S P-7 & \text { 30. } S P=R S^{2}\end{array} \)

To solve for the lengths of each side of the square \( P Q R S \) given the conditions, we will define the lengths of the sides using variables and then set up equations based on the provided relationships. 1. **Define Variables:** - Let \( a = PQ \) - Let \( b = QR \) - Let \( c = RS \) - Let \( d = SP \) 2. **Set Up Equations Based on Conditions:** - From condition 27: \( a = b \) - From condition 28: \( b = 3c \) - From condition 29: \( c = d - 7 \) - From condition 30: \( d = c^2 \) 3. **Perimeter Equation:** The perimeter of the square is given by: \[ a + b + c + d = 84 \] 4. **Substituting Variables:** Using the relationships defined: - Substitute \( b \) with \( a \) (from condition 27): \( b = a \) - Substitute \( c \) with \( \frac{b}{3} \) (from condition 28): \( c = \frac{a}{3} \) - Substitute \( d \) with \( c + 7 \) (from condition 29): \( d = \frac{a}{3} + 7 \) - Substitute \( d \) with \( c^2 \) (from condition 30): \( d = \left(\frac{a}{3}\right)^2 \) 5. **Setting Up the Perimeter Equation:** Now we can substitute these into the perimeter equation: \[ a + a + \frac{a}{3} + \left(\frac{a}{3}\right)^2 = 84 \] Simplifying this gives: \[ 2a + \frac{a}{3} + \frac{a^2}{9} = 84 \] 6. **Finding a Common Denominator:** The common denominator for the terms is 9: \[ \frac{18a}{9} + \frac{3a}{9} + \frac{a^2}{9} = 84 \] Combining the terms: \[ \frac{21a + a^2}{9} = 84 \] 7. **Multiplying Through by 9:** \[ 21a + a^2 = 756 \] Rearranging gives: \[ a^2 + 21a - 756 = 0 \] 8. **Solving the Quadratic Equation:** We can use the quadratic formula \( a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \): Here, \( A = 1, B = 21, C = -756 \): \[ a = \frac{-21 \pm \sqrt{21^2 - 4 \cdot 1 \cdot (-756)}}{2 \cdot 1} \] \[ a = \frac{-21 \pm \sqrt{441 + 3024}}{2} \] \[ a = \frac{-21 \pm \sqrt{3465}}{2} \] 9. **Calculating \( \sqrt{3465} \):** Let's calculate \( \sqrt{3465} \): \[ \sqrt{3465} \approx 58.8 \] Thus, \[ a = \frac{-21 \pm 58.8}{2} \] 10. **Finding Positive Solution:** We take the positive root: \[ a = \frac{37.8}{2} \approx 18.9 \] 11. **Finding Other Sides:** Now we can find \( b, c, d \): - \( b = a = 18.9 \) - \( c = \frac{a}{3} \approx 6.3 \) - \( d = c + 7 \approx 13.3 \) 12. **Final Lengths:** The lengths of the sides are approximately: - \( PQ \approx 18.9 \) - \( QR \approx 18.9 \) - \( RS \approx 6.3 \) - \( SP \approx 13.3 \) Thus, the lengths of each side of \( \square P Q R S \) are approximately \( 18.9, 18.9, 6.3, \) and \( 13.3 \).