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# In the diagram to the right, triangle PQR has a right angle

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In the diagram to the right, triangle PQR has a right angle [#permalink]

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29 Mar 2014, 00:17
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In the diagram to the right, triangle PQR has a right angle at Q and line segment QS is perpendicular to PR. If line segment PS has a length of 16 and line segment SR has a length of 9, what is the area of triangle PQR?

A. 72
B. 96
C. 108
D. 150
E. 200
[Reveal] Spoiler: OA

Last edited by Bunuel on 29 Mar 2014, 03:25, edited 1 time in total.
Renamed the topic and edited the question.

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Re: In the figure to the right, triangle PQR has a right angle [#permalink]

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29 Mar 2014, 01:54
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sudeeptasahu29 wrote:
Triangle PQR has a right angle at Q and the line segment QS is perpendicular to PR. If line segment PS = 16 and SR has a length of 9, what is the area of triangle PQR?

a. 72
b. 96
c. 108
d. 150
e. 200

In right triangle PQR right angled at Q, QS^2 = PS * SR --------> qs^2 = 16*9 ------> qs = 4*3 = 12

Area = $$\frac{1}{2} QS*PR$$ -----> $$\frac{1}{2} 12*25$$ ------> 150
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Re: In the figure to the right, triangle PQR has a right angle [#permalink]

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29 Mar 2014, 02:00
Dear Narenn,

Thank you for the reply. Please explain how you got this relation QS^2 = PS * SR??

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Re: In the figure to the right, triangle PQR has a right angle [#permalink]

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29 Mar 2014, 02:39
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sudeeptasahu29 wrote:
Dear Narenn,

Thank you for the reply. Please explain how you got this relation QS^2 = PS * SR??

It's a standard relationship based on similar triangle theory.

We should note that triangle PQR, triangle PSQ, and triangle QSR are similar

In similar triangles, ratios of corresponding sides are equal.

We know that triangle PSQ is similar to triangle QSR

so, $$\frac{base of PSQ}{height of PSQ} = \frac{base of QSR}{height of QSR}$$

$$\frac{QS}{PS} = \frac{RS}{QS}$$ --------> $$QS^2$$ = PS*RS

Similarly you can also prove that $$PQ^2$$= PS*PR and $$QR^2$$ = SR*PR
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Re: In the diagram to the right, triangle PQR has a right angle [#permalink]

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29 Mar 2014, 03:39
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sudeeptasahu29 wrote:

In the diagram to the right, triangle PQR has a right angle at Q and line segment QS is perpendicular to PR. If line segment PS has a length of 16 and line segment SR has a length of 9, what is the area of triangle PQR?

A. 72
B. 96
C. 108
D. 150
E. 200

Important property: the perpendicular to the hypotenuse will always divide the triangle into two triangles with the same properties as the original triangle. Check here: if-arc-pqr-above-is-a-semicircle-what-is-the-length-of-144057.html#p1154669

According to the above, triangles PQR, PSQ and QSR must be similar.

Now, since triangles PSQ and QSR are similar, the the ratio of their corresponding sides must be the same (corresponding sides are opposite equal angles): PS/QS = QS/SR --> QS^2 = PS*SR = 16*9 --> QS = 4*3 = 12.

The area of triangle PQR = 1/2*base*height = 1/2*(16+9)*12 = 150.

Hope it's clear.
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Re: In the diagram to the right, triangle PQR has a right angle [#permalink]

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13 Jun 2014, 12:49
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Hi Bunuel,

I guess you have mentioned triangle PSR in place of triangle PSQ by mistake.

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Re: In the diagram to the right, triangle PQR has a right angle [#permalink]

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14 Jun 2014, 01:28
maggie27 wrote:
Hi Bunuel,

I guess you have mentioned triangle PSR in place of triangle PSQ by mistake.

Yes, PSR is a line segment. Edited a typo. Thank you.
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Re: In the diagram to the right, triangle PQR has a right angle [#permalink]

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16 Jun 2014, 00:35
Bunuel wrote:
sudeeptasahu29 wrote:

In the diagram to the right, triangle PQR has a right angle at Q and line segment QS is perpendicular to PR. If line segment PS has a length of 16 and line segment SR has a length of 9, what is the area of triangle PQR?

A. 72
B. 96
C. 108
D. 150
E. 200

Important property: the perpendicular to the hypotenuse will always divide the triangle into two triangles with the same properties as the original triangle. Check here: if-arc-pqr-above-is-a-semicircle-what-is-the-length-of-144057.html#p1154669

According to the above, triangles PQR, PSQ and QSR must be similar.

Now, since triangles PSQ and QSR are similar, the the ratio of their corresponding sides must be the same (corresponding sides are opposite equal angles): PS/QS = QS/SR --> QS^2 = PS*SR = 16*9 --> QS = 4*3 = 12.

The area of triangle PQR = 1/2*base*height = 1/2*(16+9)*12 = 150.

Hope it's clear.

Hello Bunnel, how can we ascertain that PSQ and QSR but not PSQ and RSQ that are similar?

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Re: In the diagram to the right, triangle PQR has a right angle [#permalink]

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16 Jun 2014, 00:38
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Kconfused wrote:
Bunuel wrote:
sudeeptasahu29 wrote:

In the diagram to the right, triangle PQR has a right angle at Q and line segment QS is perpendicular to PR. If line segment PS has a length of 16 and line segment SR has a length of 9, what is the area of triangle PQR?

A. 72
B. 96
C. 108
D. 150
E. 200

Important property: the perpendicular to the hypotenuse will always divide the triangle into two triangles with the same properties as the original triangle. Check here: if-arc-pqr-above-is-a-semicircle-what-is-the-length-of-144057.html#p1154669

According to the above, triangles PQR, PSQ and QSR must be similar.

Now, since triangles PSQ and QSR are similar, the the ratio of their corresponding sides must be the same (corresponding sides are opposite equal angles): PS/QS = QS/SR --> QS^2 = PS*SR = 16*9 --> QS = 4*3 = 12.

The area of triangle PQR = 1/2*base*height = 1/2*(16+9)*12 = 150.

Hope it's clear.

Hello Bunnel, how can we ascertain that PSQ and QSR but not PSQ and RSQ that are similar?

Triangle QSR is the same as RSQ.
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Re: In the diagram to the right, triangle PQR has a right angle [#permalink]

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16 Jun 2014, 05:24
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Thanks Bunnel!
Wouldn't that mix up the corresponding sides? Obviously QSR and RSQ when matched with PSQ would result in different set of corresponding sides
My question is, how can we find the right set of corresponding sides to match up in similar triangles?
Thank you in advance! I've always had this problem with similar triangles.

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Re: In the diagram to the right, triangle PQR has a right angle [#permalink]

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18 Jun 2014, 06:03
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Re: In the diagram to the right, triangle PQR has a right angle [#permalink]

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06 Jul 2014, 10:30
Kconfused wrote:
Thanks Bunnel!
Wouldn't that mix up the corresponding sides? Obviously QSR and RSQ when matched with PSQ would result in different set of corresponding sides
My question is, how can we find the right set of corresponding sides to match up in similar triangles?
Thank you in advance! I've always had this problem with similar triangles.

i had the same issue as in finding corresponding sides in similiar triangle. i suggest you watch some videos in khan academy relating to similiar triangles. there are few videos pertaining to it .watch them they are really helpful.

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Re: In the diagram to the right, triangle PQR has a right angle [#permalink]

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17 Jul 2015, 14:09
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Re: In the diagram to the right, triangle PQR has a right angle [#permalink]

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04 Aug 2016, 22:15
Hello from the GMAT Club BumpBot!

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Re: In the diagram to the right, triangle PQR has a right angle [#permalink]

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09 Dec 2016, 02:53
Hi Bunnel,

I tried solving this problem by using the 45-45-90 triangle approach for triangle QSR !! Why is this approach incorrect ?

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Re: In the diagram to the right, triangle PQR has a right angle [#permalink]

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09 Dec 2016, 02:59
anuj11 wrote:
Hi Bunnel,

I tried solving this problem by using the 45-45-90 triangle approach for triangle QSR !! Why is this approach incorrect ?

First of all, you don't show your work at all and asks what's wrong with it. Next, why do you assume that QSR is 45-45-90 triangle?
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Re: In the diagram to the right, triangle PQR has a right angle [#permalink]

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02 Aug 2017, 21:33
Can we solve this by using the following approach??
the area of triangle PQR = area of triangle PQS+ area of QRS
But the side QS must be equal to what ?? IS that equal to PS or SR
Kindly help

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Re: In the diagram to the right, triangle PQR has a right angle   [#permalink] 02 Aug 2017, 21:33
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