September 2023
Problem of the Month

Trapezoidal Problem
Submitted by Brijesh Dave

ABCD is a trapezoid.
AB and CD are perpendicular to AD as shown in the diagram below.
Also AB + CD = BC.
The length of AD = 7 units.

What is the product of AB and CD ?

Solution:

(AB) (CD) = 49/4

Draw a line from B perpendicular to CD and call the foot of the perpendicular E.


Let x = CD and let y = AB.
So we are looking for x y.

CE = x - y

Use the Pythagorean Theorem to find CB:     (x - y)² + 7² = (BC)²

Also we know that x + y = BC (given)

Substituting, we get
(x - y)² + 7² = (x + y)²

Expanding, we get:
x² -2xy + y² + 49 = x²+ 2xy + y²
so, 4xy = 49
xy = 49/4

Click here for Carlos de Armas' excellent solution

Correctly solved by:

1. K. Sengupta	Calcutta, INDIA
2. Davit Banana	Istanbul, Turkey
3. Dr. Hari Kishan	D.N. College, Meerut, Uttar Pradesh, India
4. Rob Miles	Northbrook, Illinois, USA
5. Sudhir Bavdekar	Mumbai, India
6. Colin (Yowie) Bowey	Beechworth, Victoria, Australia
7. Carlos de Armas	Barcelona City, Spain
8. Seth Cohen	Concord, New Hampshire, USA
9. Kelly Stubblefield	Mobile, Alabama, USA

Send any comments or questions to: David Pleacher