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In ΔPQS above, if PQ =3 and PS = 4, then PR =?

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Re: In ΔPQS above, if PQ =3 and PS = 4, then PR =?  [#permalink]

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New post 22 Dec 2018, 05:43
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Hey there,

I am equating areas as it is the exact same triangle. For example ... if there was a triangle ABC whose area was known to be 30. Also, you knew that the area of the triangle was 1/2*AB*BC, we would equate the two as it is triangle ABC. SO above we are talking about triangle QPS only and hence can equate. The perpendicular from the right angle to the opposite side ( hypotenuse) can be a bisector only if it is an isosceles right triangle ( which is not the case here as it is a 3-4-5 triangle.

Is it clear now?

We are just equating two different ways to write the area of triangle QPS which is 1/2 * QP * PS and 1/2* PR * QS ...

Best,
Gladi
dave13 wrote:

Gladiator59 thank you, so are equating areas of triangles based on the rule below ? correct ?

If the measures of the corresponding sides of two triangles are proportional then the triangles are similar. Likewise if the measures of two sides in one triangle are proportional to the corresponding sides in another triangle and the including angles are congruent then the triangles are similar.

source: https://www.mathplanet.com/education/ge ... /triangles

but how can I know that PR is not a median ?

have a look here https://www.khanacademy.org/math/geomet ... equal-area

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Re: In ΔPQS above, if PQ =3 and PS = 4, then PR =?  [#permalink]

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New post 22 Dec 2018, 06:38
thanks Gladi, got it :)
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Re: In ΔPQS above, if PQ =3 and PS = 4, then PR =?  [#permalink]

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New post 28 Jun 2019, 06:57
prasannar wrote:
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In ΔPQS above, if PQ =3 and PS = 4, then PR =?

A. 9/4
B. 12/5
C. 16/5
D. 15/4
E. 20/3

Project PS Butler : Question #88


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the triangle can be figured out to be a 3:4:5 triangle..

when you see that there is a line from one of vertices to a side of the triangle (90deg) it cuts the triangle in half..
therefore,
triangle PSR is now 45 : 45 : and 90 (where line bisects the side QS)
and
QPR too, is a 45 : 45 : 90
as we know, the 45 : 45 sides will be of same length...
as QR is 5
we can say QR = RS = 2.5
2.5 is side RS
so PR will be 2.5

only answer closest to that is option B
feel free to correct my opinion
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Re: In ΔPQS above, if PQ =3 and PS = 4, then PR =?  [#permalink]

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New post 03 Jul 2019, 05:49
Shrey9 wrote:
when you see that there is a line from one of vertices to a side of the triangle (90deg) it cuts the triangle in half..


You are not cutting the triangle in half here - that would only be true if the large triangle was isosceles, and it's not. That's why your answer doesn't match any of the answer choices.
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Re: In ΔPQS above, if PQ =3 and PS = 4, then PR =?  [#permalink]

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New post 07 Jul 2019, 17:02
We we cannot solve it like this?

PRS will become a 45 - 45 - 90 triangele, so \(x\sqrt{2}= 4\) and solving it will lead us to:

PR = RS = \(2\sqrt{2}\)

What is wrong with it?

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Re: In ΔPQS above, if PQ =3 and PS = 4, then PR =?  [#permalink]

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New post 07 Jul 2019, 17:23
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jfranciscocuencag wrote:
PRS will become a 45 - 45 - 90 triangele, so \(x\sqrt{2}= 4\) and solving it will lead us to:


PRS is not a 45-45-90 triangle. The line PR would only cut the 90 degree angle at P perfectly in half if the big triangle was isosceles, and it is not isosceles.

You can see PRS can't be a 45-45-90 triangle by looking at the angles. In the big 3-4-5 triangle, the smallest angle is opposite the smallest side, so the smallest angle is at S. You'd need trigonometry to find that angle but it's definitely less than 45 degrees (it's actually roughly 37 degrees) if it's the smallest angle in a right triangle. But that angle at S is also one of the angles in triangle PRS, so PRS is not a 45-45-90 triangle.
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Re: In ΔPQS above, if PQ =3 and PS = 4, then PR =?   [#permalink] 07 Jul 2019, 17:23

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