jmc

Question

In trapezoid ABCD, AB and CD are perpendicular to AD, with AB+CD=BC, AB<CD, and AD=7. What is AB CD? Express your answer in decimal form.

Accepted Answer

Let E be the foot of the perpendicular from B to CD. Then AB = DE and BE = AD = 7. By the Pythagorean Theorem, align* AD^2 = BE^2 &= BC^2 - CE^2 &= (CD+AB)^2 - (CD - AB)^2 &=(CD+AB+CD-AB)(CD+AB-CD+AB) &=4 CD AB. align*Hence, AB CD = AD^2/4=7^2/4=49/4=12.25. FIGURE