1. Two villages A and B lie on opposite sides of a straight river in the positions shown. The perpendicular distance from A to the river is 1 km and that from B to the river is 3 km. The river is 1 km wide and village B is 3 km down the river from village A. If we build a bridge in an optimal position across and perpendicular to the river, what is the shortest distance from A to the bridge, across the bridge, and onward to B?
2. Let ABCDE be a convex pentagon with AB = BC and CD = DE. If m∠ABC = 120◦ , m∠CDE = 60◦ , and BD = 2, find the area of ABCDE.
3. Point P is inside the square ABCD such that PA = 7, PB = 5, and PC = 1. What is the area of the square?
4. Trapezoid ABCD has height 4 and DC ∥ AB. The diagonals DB and AC are perpendicular to each other, and AC = 5. What is the area of the trapezoid?
5. Triangle ABC is isosceles with AB = AC. D is the midpoint of BC. E is on AC such that DE ⊥ AC. F is the midpoint of DE. The intersection of AF and BE is G. What is m∠AGE?
6. In trapezoid ABCD with AB ∥CD, m∠D AB +m∠CB A = 90◦ , AB = 20, and CD = 8. E and F are midpoints of AB and CD, respectively. What is the length of EF?