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Hi I request you to please solve this question. Given below the solution, I couldn't get how op^2 = QS^2

In the rectangular coordinate system above, if ΔOPQ and ΔQRS have equal area, what are the coordinates of point R ?

Since ΔOPQ and ΔQRS have equal area, then \(\frac{1}{2}*OP*OQ = \frac{1}{2}*RS*QS\), which gives \(OP*OQ = RS*QS\).

(1) The coordinates of point P are (0,12). This implies that OP = 12. Not sufficient.

(2) OP = OQ and QS = RS. No values are given, so this statement is clearly insufficient. But from this statement we get that \(OP^2 = RS^2\) (from \(OP*OQ = RS*QS\)), which gives OP = RS. So, OP = OQ = QS = RS. Basically we get that ΔOPQ and ΔQRS are congruent, similar triangles.

(1)+(2) \(OP^2 = RS^2=12^2\). So, \(OP = OQ = QS = RS=12\). Thus the coordinates of R are (24, 12). Sufficient.

Re: In the rectangular coordinate system above, if ΔOPQ and ΔQRS have equa [#permalink]

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15 Jul 2017, 04:50

For statement 1, can't a pythagorean triplet apply? That way A is also sufficient. (12,13,5) B) is sufficient on its own too, giving us triangle 1 with (0,0) (0,12) and since two sides are equal, (12,0)for the coordinate Q. And since we have (12,0) we get 24,0 and the last coordinate of R

Could someone explain why D is wrong? And what is the error in the above logic?

For statement 1, can't a pythagorean triplet apply? That way A is also sufficient. (12,13,5) B) is sufficient on its own too, giving us triangle 1 with (0,0) (0,12) and since two sides are equal, (12,0)for the coordinate Q. And since we have (12,0) we get 24,0 and the last coordinate of R

Could someone explain why D is wrong? And what is the error in the above logic?

For (1) infinitely many other cases are possible. For example, (12, 1, \(\sqrt{145}\)), (12, 2, \(\sqrt{148}\)), (12, 1.5, \(\sqrt{146.25}\)), ... Generally knowing only one side of a triangle is not enough to find other sides.

For (2) no values are given. You cannot use info from one statement when solving another.
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