1. **Problem statement:** Find the length of side $x = BC$ in the quadrilateral with given sides and angles. 2. **Given:** - $AB = 5.6$ cm - $BC = x$ cm (unknown) - $CD = 6.7$ cm - $AD = 9.4$ cm - $BD = 8.9$ cm - $\angle ABC = 32^\circ$ - $\angle BAD = 72^\circ$ - $\angle ADC = y^\circ$ (unknown) 3. **Approach:** Use the Law of Cosines in triangles $ABD$ and $BCD$ to find $x$. 4. **Step 1: Find $BD$ using triangle $ABD$:** - We know $AB$, $AD$, and $\angle BAD$. - Law of Cosines: $$BD^2 = AB^2 + AD^2 - 2 \times AB \times AD \times \cos(72^\circ)$$ - Substitute values: $$BD^2 = 5.6^2 + 9.4^2 - 2 \times 5.6 \times 9.4 \times \cos(72^\circ)$$ - Calculate: $$5.6^2 = 31.36$$ $$9.4^2 = 88.36$$ $$\cos(72^\circ) \approx 0.3090$$ - So, $$BD^2 = 31.36 + 88.36 - 2 \times 5.6 \times 9.4 \times 0.3090$$ $$BD^2 = 119.72 - 32.54 = 87.18$$ - Therefore, $$BD = \sqrt{87.18} \approx 9.34 \text{ cm}$$ 5. **Step 2: Check given $BD = 8.9$ cm:** - The problem states $BD = 8.9$ cm, which differs from our calculation. - We will use the given $BD = 8.9$ cm for further calculations. 6. **Step 3: Use triangle $ABC$ to find $x$:** - In triangle $ABC$, sides $AB = 5.6$ cm, $BC = x$, and angle $ABC = 32^\circ$. - We need more information to apply Law of Cosines or Sines. 7. **Step 4: Use triangle $BCD$ with known sides $CD = 6.7$ cm, $BD = 8.9$ cm, and angle $BCD = y^\circ$ (unknown):** - Without $y$, we cannot directly solve for $x$. 8. **Step 5: Use Law of Cosines in triangle $ABC$ with side $AC$ unknown:** - Since $AC$ is not given, and $y$ is unknown, we cannot solve directly. 9. **Step 6: Use Law of Cosines in triangle $BCD$ to express $x$ in terms of $y$:** - $$BD^2 = BC^2 + CD^2 - 2 \times BC \times CD \times \cos(y^\circ)$$ - Substitute known values: $$8.9^2 = x^2 + 6.7^2 - 2 \times x \times 6.7 \times \cos(y^\circ)$$ - Simplify: $$79.21 = x^2 + 44.89 - 13.4 x \cos(y^\circ)$$ - Rearranged: $$x^2 - 13.4 x \cos(y^\circ) + 44.89 - 79.21 = 0$$ $$x^2 - 13.4 x \cos(y^\circ) - 34.32 = 0$$ 10. **Step 7: Use Law of Cosines in triangle $ABD$ to find $BD$ (already given as 8.9 cm) and check consistency. Since $y$ is unknown, we cannot solve for $x$ exactly without $y$ or additional information.** **Conclusion:** The problem as stated lacks sufficient information to find $x$ exactly because angle $y$ is unknown and necessary for the Law of Cosines in triangle $BCD$. If $y$ is provided, $x$ can be found by solving the quadratic equation in step 9. **Final answer:** Cannot determine $x$ without angle $y$.