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Director  Joined: 05 Jan 2008
Posts: 542
In ΔPQS above, if PQ =3 and PS = 4, then PR =?  [#permalink]

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Difficulty:   35% (medium)

Question Stats: 71% (01:59) correct 29% (02:41) wrong based on 475 sessions

HideShow timer Statistics 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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Math Expert V
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Re: In ΔPQS above, if PQ =3 and PS = 4, then PR =?  [#permalink]

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5
9
carcass wrote: In the figure, if the length of PQ is 4 and the length of PS is 3, what is the length of line PR?

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

Hypotenuse $$QS=\sqrt{3^2+4^2}=5$$;

The area of the triangle PQS is $$area=\frac{1}{2}*PQ*PS=6$$;

But the are can be found in another way too: $$area=\frac{1}{2}*PR*QS=\frac{5}{2}*PR$$ --> $$\frac{5}{2}*PR=6$$ --> $$PR=\frac{12}{5}$$.

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

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prasannar wrote:
In ΔPQS attached, if PQ =3 and PS = 4, then PR=?

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

its just equating the ares

1/2(pq*ps)=1/2(qs*pr)

1/2(3*4)=1/2 (5*pr)

pr=12/5
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Manager  Joined: 24 Apr 2008
Posts: 128
Re: In ΔPQS above, if PQ =3 and PS = 4, then PR =?  [#permalink]

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Let QR=x and PR=h

x^2 +h^2 = 9

5-x)^2 + h^2 = 16

Solving for x = 9/5 and h=12/5
Director  Joined: 23 Sep 2007
Posts: 610
Re: In ΔPQS above, if PQ =3 and PS = 4, then PR =?  [#permalink]

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1
B

3-4-5 triangle

let qr = x
then rs = 5-x

let pr = h

2 equations:

3^2 = h^2 + x^2
and
4^2 = h^2 + (5-x)^2

expand and substract them to get x = 9/5

substitute 9/5 into the first equation (or second, your choice)
to get
9 = h^2 + 81/25

solve the equation to get h = 12/5
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Re: In ΔPQS above, if PQ =3 and PS = 4, then PR =?  [#permalink]

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4
1
This is quite a well-known diagram in mathematics: it can be used to prove the Pythagorean Theorem (using a and b for the two legs, and not 3 and 4). There are (at least) three entirely different ways to solve this problem, two of which were described above. Call the length we're looking for 'd':

-The area of the triangle must be the same no matter which base you choose. Thus 3*4/2 = 5*d/2 --> d = 12/5.

-Let QR = c; then QS = 5-c. We have two right angled triangles, and can use Pythagoras to set up two equations, two unknowns (c and d; this is the most time-consuming approach).

-The approach not mentioned above: notice that the two smaller triangles in the diagram are each similar to the 3-4-5 triangle. Because PQR is similar to QSP, we have d/3 = 4/5 --> d = 12/5.

It's the similarity of the triangles that can let you prove Pythagoras, incidentally.
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Re: In ΔPQS above, if PQ =3 and PS = 4, then PR =?  [#permalink]

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4
PQ =3 and PS = 4,
So QS = 5

area of the triangle = 1/2 * PQ * PS = 1/2*QS*PR

OR, 1/2 * 3 * 4 = 1/2 * QS * PR
OR PR = 12/5
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Re: In ΔPQS above, if PQ =3 and PS = 4, then PR =?  [#permalink]

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Attachment: Triangle.jpg [ 21.24 KiB | Viewed 24868 times ]
In the figure, if the length of PQ is 4 and the length of PS is 3, what is the length of line PR?

(A) 9/4
(B) 12/5
(C) 16/5
(D) 15/4
(E) 20/3
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Re: In ΔPQS above, if PQ =3 and PS = 4, then PR =?  [#permalink]

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Be patient. I have not fully understood

Why 5/2 * PQ...should be QS ??

Secondly why my reasoning do not lead me to the answer and is flawed, completely wrong ???

We have a right triangle PQS where angle P is 90. Also two sides PS=3 and PQ= 4 so from this we have a 30-60-90 triangle.

So, if we look at angle R is 90, from this we can see that angle P is shared between PQS and PRS so P for PRS should be 60 (likewise angle P for triangle PRQ should be 30, so angle P is 60+30=90).

So, we have: for triangle PRS angle R is 90 (PS is the hypotenuse), angle P is 60 (RS long leg) and angle S is 30 (short leg PR).

If PS is 3 (hypotenuse opposite 90 angle) PR should be 1.5 (short leg opposite angle S that is 30). This based on ratio 30:60:90.

I know that it does not hold anywater, but is useful to understand Thanks
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Re: In ΔPQS above, if PQ =3 and PS = 4, then PR =?  [#permalink]

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carcass wrote:
Be patient. I have not fully understood

Why 5/2 * PQ...should be QS ??

Secondly why my reasoning do not lead me to the answer and is flawed, completely wrong ???

We have a right triangle PQS where angle P is 90. Also two sides PS=3 and PQ= 4 so from this we have a 30-60-90 triangle.

So, if we look at angle R is 90, from this we can see that angle P is shared between PQS and PRS so P for PRS should be 60 (likewise angle P for triangle PRQ should be 30, so angle P is 60+30=90).

So, we have: for triangle PRS angle R is 90 (PS is the hypotenuse), angle P is 60 (RS long leg) and angle S is 30 (short leg PR).

If PS is 3 (hypotenuse opposite 90 angle) PR should be 1.5 (short leg opposite angle S that is 30). This based on ratio 30:60:90.

I know that it does not hold anywater, but is useful to understand Thanks

First of all 5/2 * PQ does not equal to QS. We equate the areas, which can be found in two ways:
1. 1/2*Leg1*Leg2 --> $$area=\frac{1}{2}*PQ*PS=6$$;
2. 1/2*Perpendicular to hypotenuse*Hypotenuse --> $$area=\frac{1}{2}*PR*QS=\frac{5}{2}*PR$$ (since hypotenuse QS=5);

Now, equate the areas: $$6=\frac{5}{2}*PR$$ --> $$PR=\frac{12}{5}$$.

Next, in a right triangle where the angles are 30°, 60°, and 90° the sides are always in the ratio $$1 : \sqrt{3}: 2$$. In PQS sides PQ and PS are NOT in the ratio $$1 : \sqrt{3}$$, so PQS is not a 30°, 60°, and 90° right triangle. I think you are mixing 3-4-5 Pythagorean Triples triangle with 30°-60°-90° triangle.

Hope it's clear.
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Re: In ΔPQS above, if PQ =3 and PS = 4, then PR =?  [#permalink]

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It does make sense.

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

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5
As mentioned by Bunuel, there are other methods as well to solve this problem

One of the method is Similarity
Triangle QPS is similar to PRS (because one common side & angle is 3 & 90 degree)
So,
QS/QP = PS/PR
5/4 = 3/x
x = 12/5

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

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prasannar wrote:
Attachment:
Diag-1.JPG
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

Two approaches. the area is 6. so area of pqr + area of prs = 6. solution will result in 12/5. B

second: testing answer choices. if assume that line(pr) is an answer choice and try to rearrange to get line(rs), then you discover line rs is squareroot of some negative numbers in all the options except B
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Re: In ΔPQS above, if PQ =3 and PS = 4, then PR =?  [#permalink]

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prasannar wrote:
Attachment:
Diag-1.JPG
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

QS will be 5 being the hypotenuse of 90 degree triangle.

As we can calculate the area by two ways here:

1/2 * PS * PQ = 1/2 * PR * QS

4 * 3 = PR * 5

PR = 12/5
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Re: In ΔPQS above, if PQ =3 and PS = 4, then PR =?  [#permalink]

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Very simple hypotenuse will be 5 as right angle triangle (3,4,5) . A perpendicular drawn to hypotenuse will bisect it , that means 5/2 or 2.5 ; 12/5 nearest option.
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Re: In ΔPQS above, if PQ =3 and PS = 4, then PR =?  [#permalink]

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Let QR=x and RS=y.
Area of QPS= 1/2*4*3 = 6
We know, Area of QPS = Area of QRP + Area of PRQ --> (1)
Let PR=a.
From Pythagoras Theorem, QS=5. ==> x+y=5
In (1), 6 = 1/2*y*a +1/2*x*a ==> 1/2(ay + ax) ==> a(x+y) =12
As x+y=5, a*5=12 ==> a=12/5 = PR
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Re: In ΔPQS above, if PQ =3 and PS = 4, then PR =?  [#permalink]

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1
PQS is a 3-4-5 triangle, so QS = 5. PR is an altitude of PQS. The altitude of a right triangle forms two more triangles that are similar to the original and each other. So pick any of the smaller triangles and draw it with the same orientation as PQS. Then we can set up a proportion since corresponding sides of similar triangles are proportional. From that we get RP/PS = QP/QS --> RP/4 = 3/5 --> RP =12/5 B)
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In ΔPQS above, if PQ =3 and PS = 4, then PR =?  [#permalink]

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Bunuel wrote:
carcass wrote: In the figure, if the length of PQ is 4 and the length of PS is 3, what is the length of line PR?

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

Hypotenuse $$QS=\sqrt{3^2+4^2}=5$$;

The area of the triangle PQS is $$area=\frac{1}{2}*PQ*PS=6$$;

But the are can be found in another way too: $$area=\frac{1}{2}*PR*QS=\frac{5}{2}*PR$$ --> $$\frac{5}{2}*PR=6$$ --> $$PR=\frac{12}{5}$$.

Bunuel, Gladiator59, chetan2u i have questions Area of which traigle do you find through this $$area=\frac{1}{2}*PR*QS=\frac{5}{2}*PR$$

I know hypotenuse of trangle QPS is 5, and 5/2 means you are dividing hypotenuse by 2 since PR bisects triangle QPS

can please help me to understand the logic when you equate length of shorter leg 2.5 to area of triangle QPS $$\frac{5}{2}*PR=6$$

Also does PR bisect triangle QPS ? i think it bisects, because point R is 90 degrees. If my statement is true Yes than shorter leg (RS) of traingle PSR will be (5/2) =2.5

So if hypotenuse (PS) of triangle PSR is 4 and shorter leg (RS) is 2.5 find longer leg PR
we can use a pythogorean formula

$$X^2+2.5^2=4^2$$

$$x^2+6.25= 16$$

$$x^2=16 - 6.25$$

$$x^2=9.75$$ on taking square root

$$x = 3.12$$(approximatif)

Whats wring with my reasoning Senior PS Moderator D
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GMAT 1: 740 Q50 V40 GMAT 2: 770 Q51 V42 Re: In ΔPQS above, if PQ =3 and PS = 4, then PR =?  [#permalink]

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Please find my responses in-line in red.

dave13 wrote:

Area of which traigle do you find through this $$area=\frac{1}{2}*PR*QS=\frac{5}{2}*PR$$ Area of the big triangle - as PR is perpendicular to the hypotenuse and the sides are perpendicular to each other - we can find the area by applying the formula to either PR and QS or PQ and PS

I know hypotenuse of trangle QPS is 5, and 5/2 means you are dividing hypotenuse by 2 since PR bisects triangle QPS - Just applying the formula for area

can please help me to understand the logic when you equate length of shorter leg 2.5 to area of triangle QPS $$\frac{5}{2}*PR=6$$ Equating the areas as it is the SAME TRIANGLE!!

Also does PR bisect triangle QPS ? i think it bisects, because point R is 90 degrees. If my statement is true Yes than shorter leg (RS) of traingle PSR will be (5/2) =2.5 Does not bisect
So if hypotenuse (PS) of triangle PSR is 4 and shorter leg (RS) is 2.5 find longer leg PR
we can use a pythogorean formula

$$X^2+2.5^2=4^2$$

$$x^2+6.25= 16$$

$$x^2=16 - 6.25$$

$$x^2=9.75$$ on taking square root

$$x = 3.12$$(approximatif)

LAST PART ( IN MAROON) IS WRONG AS PR DOESNT BISECT

Whats wring with my reasoning Just equate the area by two different methods - this is the fastest solution. There is another solution by similar triangles - can you find it?  Regards,
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In ΔPQS above, if PQ =3 and PS = 4, then PR =?  [#permalink]

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Just equate the area by two different methods - this is the fastest solution. There is another solution by similar triangles - can you find it?  Regards,

Gladiator59 thank you, so we 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 In ΔPQS above, if PQ =3 and PS = 4, then PR =?   [#permalink] 22 Dec 2018, 05:37

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