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

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Math Expert
Joined: 02 Sep 2009
Posts: 56304
Are rectangles 1 and 2 equal in area? (1) Angle x measures 45°. (2) O  [#permalink]

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18 Sep 2018, 22:00
1
5
00:00

Difficulty:

95% (hard)

Question Stats:

33% (01:55) correct 68% (01:55) wrong based on 49 sessions

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Are rectangles 1 and 2 equal in area?

(1) Angle x measures 45°.
(2) OP = PQ

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GMATH Teacher
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Re: Are rectangles 1 and 2 equal in area? (1) Angle x measures 45°. (2) O  [#permalink]

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25 Sep 2018, 15:56
Bunuel wrote:

Are rectangles 1 and 2 equal in area?

(1) Angle x measures 45°.
(2) OP = PQ

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.
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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
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Re: Are rectangles 1 and 2 equal in area? (1) Angle x measures 45°. (2) O   [#permalink] 25 Sep 2018, 15:56
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