1. **Problem statement:** Given quadrilateral ABCD with AB = 9, BC = 10, AB ∥ DC, angle B = 63°, and a perpendicular height from C to AD with foot length 4, find the perimeter of ABCD. 2. **Key facts and formulas:** - Since AB ∥ DC, ABCD is a trapezoid. - The perimeter is the sum of all sides: $P = AB + BC + CD + DA$. - Use trigonometry to find lengths CD and DA. 3. **Find height (h) from C to AD:** Given the perpendicular from C to AD, the height $h = 4$. 4. **Find length CD:** Since the perpendicular from C to AD is height $h=4$, and BC = 10 is horizontal, triangle BCD is right-angled at the foot of the perpendicular. 5. **Find length DA:** Use angle B = 63° and AB = 9 to find vertical and horizontal components of AD. 6. **Calculate AD components:** - Vertical component of AB: $AB_y = AB \sin 63^\circ = 9 \times \sin 63^\circ$ - Horizontal component of AB: $AB_x = 9 \times \cos 63^\circ$ 7. **Since AB ∥ DC, length DC = AB = 9.** 8. **Calculate DA:** - Horizontal length from foot of perpendicular to D is 4 (given). - Total horizontal length AD = $AB_x + 4$ - Vertical length AD = $AB_y$ - Length $DA = \sqrt{(AB_x + 4)^2 + (AB_y)^2}$ 9. **Calculate values:** - $\sin 63^\circ \approx 0.8910$ - $\cos 63^\circ \approx 0.4540$ - $AB_y = 9 \times 0.8910 = 8.019$ - $AB_x = 9 \times 0.4540 = 4.086$ - $DA = \sqrt{(4.086 + 4)^2 + (8.019)^2} = \sqrt{(8.086)^2 + (8.019)^2}$ - $DA = \sqrt{65.38 + 64.30} = \sqrt{129.68} \approx 11.39$ 10. **Calculate perimeter:** - $P = AB + BC + CD + DA = 9 + 10 + 9 + 11.39 = 39.39$ **Final answer:** $$\boxed{39.39}$$