Trapezoid Ef Area Ad8679

1. **State the problem:** We have trapezoid ABCD with AB parallel to CD, AB = 12 cm, CD = 18 cm, and \(\angle B = 124^\circ\). Points E and F lie on AD and BC respectively, with EF parallel to AB and CD. We need to calculate the length of EF and the area of trapezoid \(\angle C\) (likely meaning the area of trapezoid EFCB). 2. **Recall trapezoid properties:** In trapezoids, segments parallel to the bases and connecting the legs create smaller trapezoids similar to the original. 3. **Calculate EF:** Since EF is parallel to AB and CD, EF is a segment between the legs proportional to the bases. 4. **Use the formula for the length of a segment parallel to the bases in a trapezoid:** If EF is drawn parallel to AB and CD, then $$EF = AB + k(CD - AB)$$ where \(k\) is the ratio of the distance from AB to EF over the height of the trapezoid. 5. **Calculate the height and other dimensions:** We need to find the height \(h\) of trapezoid ABCD. 6. **Calculate height using angle B:** Drop perpendiculars from B and C to line AD or extend AB and CD to find height. 7. **Calculate height \(h\):** Using \(\angle B = 124^\circ\), the height is $$h = AB \times \sin(180^\circ - 124^\circ) = 12 \times \sin(56^\circ)$$ Calculate \(\sin(56^\circ) \approx 0.829\), so $$h \approx 12 \times 0.829 = 9.948 \text{ cm}$$ 8. **Calculate EF length:** Since EF is parallel and divides the trapezoid proportionally, if EF is the mid-segment, $$EF = \frac{AB + CD}{2} = \frac{12 + 18}{2} = 15 \text{ cm}$$ 9. **Calculate area of trapezoid EFCB:** Assuming EFCB is the smaller trapezoid formed by EF and CD, Area formula: $$\text{Area} = \frac{(EF + CD)}{2} \times h'$$ where \(h'\) is the height of trapezoid EFCB. 10. **Calculate height \(h'\):** If EF is the mid-segment, then \(h' = \frac{h}{2} = \frac{9.948}{2} = 4.974 \text{ cm}\) 11. **Calculate area:** $$\text{Area} = \frac{15 + 18}{2} \times 4.974 = \frac{33}{2} \times 4.974 = 16.5 \times 4.974 = 82.041 \text{ cm}^2$$ **Final answers:** - Length of EF is approximately \(15\) cm. - Area of trapezoid EFCB is approximately \(82.04\) cm\(^2\).