Question

An archeologist discovered a broken piece of what appeared to be a plate. She measured chord AC from the edge of the break to be 8 inches. She also measured the perpendicular distance from the edge of the plate to the chord, BE to be 2.5 inches. What is the diameter of the plate?

Answer 1

we know that

AE*EC=DE*EB-----------> String theorem

AC=8 in----------> AE=AC/2----------> AE=4 in
EC=AE-----------> EC=4 in
BE=2.5 in
DE=?
DE=(AE*EC)/EB----------> DE=(4*4)/2.5---------> DE=6.4 in

then 

diameter=DE+EB--------> 6.4+2.5=8.9 in

the answer is
the diameter is 8.9 in