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705-805 Level|   Geometry|      
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ChandlerBong
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PQR is an isosceles triangle and PR is greater than PQ. 29 Apr 2023, 07:17
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C
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85% (hard)

Question Stats:

33% (03:07) correct 67% (02:32) wrong based on 27 sessions

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PQR is an isosceles triangle and PR is greater than PQ. If the area of the triangle is 18 units, what is the positive difference between the unequal sides of the triangle?

(A) PR = \(\sqrt{2}\) PQ

(B) Angle P = Angle R
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C
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ChandlerBong
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PQR is an isosceles triangle and PR is greater than PQ. 29 Apr 2023, 08:44
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ChandlerBong
PQR is an isosceles triangle and PR is greater than PQ. If the area of the triangle is 18 units, what is the positive difference between the unequal sides of the triangle?

(A) PR = \(\sqrt{2}\) PQ

(B) Angle P = Angle R
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Bunuel KarishmaB ThatDudeKnows chetan2u


Hi Experts,

Can you please share your approach to this question?

Thanks in advance.

Regards.
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ThatDudeKnows
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PQR is an isosceles triangle and PR is greater than PQ. 29 Apr 2023, 09:21
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ChandlerBong
PQR is an isosceles triangle and PR is greater than PQ. If the area of the triangle is 18 units, what is the positive difference between the unequal sides of the triangle?

(A) PR = \(\sqrt{2}\) PQ

(B) Angle P = Angle R
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Statement I alone:
QR could be equal to either PQ or PR, but we don't know which, so there are two possible triangles, each of which would have its own solution to the question. Does Statement I alone give us enough information to answer the question? No. BCE.

Statement II alone:
We know that QR = PQ, but we don't know whether the triangle is very fat and very short or if the base is just a little wider than the other two sides. Does Statement II alone give us enough information to answer the question? No. CE.

Both together:
We can draw a triangle with two sides of length x and one side of length \(\sqrt{2}\)x. We can imagine inflating or deflating this figure until it has an area of 18. We COULD figure out the dimensions and angles...but we DO NOT HAVE TO. Do we have enough information to answer the question? Yes. C.
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PQR is an isosceles triangle and PR is greater than PQ. 29 Apr 2023, 23:44
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I didn't get it.
AN isosceles triangle has two sides equal and one side that is either bigger or smaller.
(I) Gives the two different sides. It should be sufficient to answer the difference between two unequal sides.
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PQR is an isosceles triangle and PR is greater than PQ. 02 May 2023, 04:38
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tsraeroshad
I didn't get it.
AN isosceles triangle has two sides equal and one side that is either bigger or smaller.
(I) Gives the two different sides. It should be sufficient to answer the difference between two unequal sides.
Show more

How do you know which side it is equal to?

PR, QR and PQ are the three sides.
PR > PQ so these are not the equal sides. So QR is one of the equal sides. But which side is it equal to? Is QR = PR or is QR = PQ? We don't know.

Using statement 1, we know that we can get two isosceles triangles:

If QR = PQ, then the triangle has sides in the ratio 1: 1: sqrt(2) (a right isosceles triangle)

If QR = PR, then the triangle has sides in the ratio 1: sqrt(2) : sqrt(2) (NOT a right triangle, but nevertheless isosceles)

They are playing on your familiarity with a 45-45-90 triangle and hoping that you will jump to that and forget that another case is possible.

Statement 2 tells us that angle P = angle R so QR = PQ and hence the triangle has sides 1: 1: sqrt(2) and area of (1/2)*leg*leg = 18 so each leg length is 6.
Length of hypotenuse then is 6*sqrt(2) and the difference between them is 6*sqrt(2) - 6

We don't need to do all this calculation at the end.

Answer (C)
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