Are rectangles 1 and 2 equal in area? (1) Angle x measures 45°. (2) O

First of all, be sure you (the reader) can justify all angles presented in the figure below (pre-statements).



\(?\,\,\,:\,\,\,\,OT \cdot VT\,\,\mathop = \limits^? \,\,\,QT \cdot ZT\)

(1) Sufficient: note that x = 45 degrees implies:

(a) triangles OPT and TPQ are congruent (both are right and isosceles, with PT a common leg), hence OT = TQ
(b) triangles TVS and SZT are congruent (both are right and isosceles, with TS a common hypotenuse), hence VT = ZS = ZT

Conclusion: (OT times VT) equals (TQ times ZT) , exactly our FOCUS. (The answer is in the affirmative.)


(2) Sufficient: OP = PQ (and right angle OPT) implies line SP is the perpendicular bisector of line segment OQ , hence OT = QT.
From this fact, we again may conclude that triangles OPT and TPQ are (not only similar but also) congruent, hence x=90-x and x = 45 degrees.

Conclusion: (2) implies (1), hence (2) is also sufficient (alone).


The correct answer is (D), indeed.


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.