12. In and BC=20. Points D,E, and F are on sides , and , respectively, such that and are parallel to and , respectively. What is the perimeter of parallelogram ADEF?

Solution: Perimeter = 2(AD + AF). But AD = EF (since ABCD is a parallelogram). Hence perimeter = 2(AF + EF). Now ABC is isosceles (AB = AC = 28). Thus angle B = angle C. But EF is parallel to AB. Thus angle FEC = angle B which in turn is equal to angle C. Hence triangle CEF is isosceles. Thus EF = CF. Perimeter = 2(AF + EF) = 2(AF + EF) =2AC = = 56.