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# In the figure given below, the length of PQ is 12 and the length of PR

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Math Expert
Joined: 02 Sep 2009
Posts: 49271
In the figure given below, the length of PQ is 12 and the length of PR  [#permalink]

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16 Apr 2015, 05:11
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Question Stats:

71% (02:36) correct 29% (03:09) wrong based on 123 sessions

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In the figure given below, the length of PQ is 12 and the length of PR is 15. The area of right triangle STU is equal to the area of the shaded region. If the ratio of the length of ST to the length of TU is equal to the ratio of the length of PQ to the length of QR, what is the length of TU?

(A) (9√2)/4
(B) 9/2
(C) (9√2)/2
(D) 6√2
(E) 12

Kudos for a correct solution.

Attachment:

Ques3.jpg [ 8.68 KiB | Viewed 5458 times ]

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Re: In the figure given below, the length of PQ is 12 and the length of PR  [#permalink]

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16 Apr 2015, 06:07
1
Bunuel wrote:
Attachment:
Ques3.jpg
In the figure given below, the length of PQ is 12 and the length of PR is 15. The area of right triangle STU is equal to the area of the shaded region. If the ratio of the length of ST to the length of TU is equal to the ratio of the length of PQ to the length of QR, what is the length of TU?

(A) (9√2)/4
(B) 9/2
(C) (9√2)/2
(D) 6√2
(E) 12

Kudos for a correct solution.

Hello,

By pythagoras theorem, Side QR = 9

Now area of Triangle PQR = 1/2 * 12 * 9 = 54.

We also know that,
Area of Triangle PQR - Area of shaded region = Area of Triangle STU.
Therefore Area of Triangle PQR = Area of Triangle STU + Area of Shaded Region
So, Area of Triangle PQR = 2 (Area of Triangle STU) (Since Area of Triangle STU = Area of shaded region)
So Area of Triangle STU = 54/2 = 27.

Side PQ : QR = Side ST : TU
12 : 9 = 4x : 3X

Now A(triangle STU) = 1/2 * 4x * 3x = 27

Solving X = 3/√2
Side TU = 3 * 3/√2= 9/√2 = 9√2/2

So the ans is C
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Re: In the figure given below, the length of PQ is 12 and the length of PR  [#permalink]

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16 Apr 2015, 09:56
Bunuel wrote:

In the figure given below, the length of PQ is 12 and the length of PR is 15. The area of right triangle STU is equal to the area of the shaded region. If the ratio of the length of ST to the length of TU is equal to the ratio of the length of PQ to the length of QR, what is the length of TU?

(A) (9√2)/4
(B) 9/2
(C) (9√2)/2
(D) 6√2
(E) 12

Kudos for a correct solution.

Attachment:
Ques3.jpg

144+QR^2=225
QR^2=81
QR=9

Area of Triangle PQR=1/2*12*9=54

Half of 54 is the area of STU
ST*TU=54...(1)

$$\frac{ST}{TU}$$=$$\frac{12}{9}$$

$$\frac{ST}{TU}$$=$$\frac{4}{3}$$

ST=$$\frac{4TU}{3}$$...(2)

Replacing (2) in (1)

$$\frac{4}{3}$$ $$TU^2$$=54
$$TU^2$$=$$\frac{81}{2}$$
TU=$$\frac{9}{\sqrt{2}}$$

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Posts: 262
Re: In the figure given below, the length of PQ is 12 and the length of PR  [#permalink]

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16 Apr 2015, 10:03
yes, Plus one for C. Straight forwrad question.

Bunuel wrote:

In the figure given below, the length of PQ is 12 and the length of PR is 15. The area of right triangle STU is equal to the area of the shaded region. If the ratio of the length of ST to the length of TU is equal to the ratio of the length of PQ to the length of QR, what is the length of TU?

(A) (9√2)/4
(B) 9/2
(C) (9√2)/2
(D) 6√2
(E) 12

Kudos for a correct solution.

Attachment:
Ques3.jpg

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Kudos to you, for helping me with some KUDOS.

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Re: In the figure given below, the length of PQ is 12 and the length of PR  [#permalink]

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16 Apr 2015, 10:15
1
C for me too..
PQR : 345 rt triangle. Thus QR = 9.
Given , ST/TU = PQ/QR = 12/9 = 4/3
=> ST = 4*TU/3

Now, area PQR = 2*area(STU)
=> 12*9/2 = ST*TU = 4*TU^2/3
=> TU^2 = 81/2
=> TU = 9/sqrt(2)

Thus ans C.
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Kudos!!

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Re: In the figure given below, the length of PQ is 12 and the length of PR  [#permalink]

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18 Apr 2015, 22:00
Given PQ = 12, PR = 15, area of right triangle STU is equal to the area of the shaded region
then Area of $$\triangle STU\,=\,\frac{1}{2}$$ Area of $$\triangle PQR$$

$$\triangle PQR$$ is a right triangle
$$QR^2\,=\,PR^2\,-\,PQ^2$$
$$QR^2\,=\,15^2\,-\,12^2$$
$$QR\,=\,9$$
Area of $$\triangle PQR\,=\,\frac{1}{2}*12*9$$$$\,=\,54$$
So
Area of $$\triangle STU\,=\,27$$

Given $$\frac{ST}{TU}\,=\,\frac{PQ}{QR}$$;$$\triangle PQR\,$$and$$\,\triangle STU$$ are similar
$$(Ratio\,of\,Sides)^2\,=\,(Ratio\,of\,Areas)$$
$$\frac{TU^2}{QR^2}\,=\,\frac{Area\,of\,\triangle STU}{Area\,of\,\triangle PQR}$$
$$TU^2\,=\,\frac{81*27}{54}$$
$$TU^2\,=\,\frac{81}{2}$$
$$TU\,=\,\frac{9}{\sqrt{2}}\,=\,\frac{9\sqrt{2}}{2}$$

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Re: In the figure given below, the length of PQ is 12 and the length of PR  [#permalink]

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19 Apr 2015, 03:32
Bunuel wrote:

In the figure given below, the length of PQ is 12 and the length of PR is 15. The area of right triangle STU is equal to the area of the shaded region. If the ratio of the length of ST to the length of TU is equal to the ratio of the length of PQ to the length of QR, what is the length of TU?

(A) (9√2)/4
(B) 9/2
(C) (9√2)/2
(D) 6√2
(E) 12

Kudos for a correct solution.

Attachment:
Ques3.jpg

QR=\sqrt{225-144}=9
Area of triangle PQR=1/2*9*12=54
Therefore, Area of each of the two triangles=27
And, 1/2*ST*TU=27
ST*TU=54
But ST/TU=PQ/QR=12/9=4:3
Therefore,4TU*TU/3=54
TU=9\sqrt{2}/2
Math Expert
Joined: 02 Sep 2009
Posts: 49271
Re: In the figure given below, the length of PQ is 12 and the length of PR  [#permalink]

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20 Apr 2015, 05:20
Bunuel wrote:

In the figure given below, the length of PQ is 12 and the length of PR is 15. The area of right triangle STU is equal to the area of the shaded region. If the ratio of the length of ST to the length of TU is equal to the ratio of the length of PQ to the length of QR, what is the length of TU?

(A) (9√2)/4
(B) 9/2
(C) (9√2)/2
(D) 6√2
(E) 12

Kudos for a correct solution.

Attachment:
Ques3.jpg

VERITAS PREP OFFICIAL SOLUTION

The information given in the question seems to overwhelm us but let’s take it a bit at a time.

“length of PQ is 12 and the length of PR is 15”

PQR is a right triangle such that PQ = 12 and PR = 15. So PQ:PR = 4:5. Recall the 3-4-5 triplet. A multiple triplet of 3-4-5 is 9-12-15. This means QR = 9.

“ratio of the length of ST to the length of TU is equal to the ratio of the length of PQ to the length of QR”

ST/TU = PQ/QR

The ratio of two sides of PQR is equal to the ratio of two sides of STU and the included angle between the sides is same ( = 90). Using SAS, triangles PQR and STU are similar.

“The area of right triangle STU is equal to the area of the shaded region”

Area of triangle PQR = Area of triangle STU + Area of Shaded Region

Since area of triangle STU = area of shaded region, (area of triangle PQR) = 2*(area of triangle STU)

In similar triangles, if the sides are in the ratio k, the areas of the triangles are in the ratio k^2. If the ratio of the areas is given as 2 (i.e. k^2 is 2), the sides must be in the ratio √2 (i.e. k must be √2).

Since QR = 9, TU must be 9/√2. But there is no 9/√2 in the options – in the options the denominators are rationalized. TU = 9/√2 = (9*√2)/(√2*√2) = (9*√2)/2.

The question could take a long time if we do not remember the Pythagorean triplets and the area of similar triangles property.

Takeaways:

Pythagorean triplets you should know: (3, 4, 5), (5, 12, 13) and (8, 15, 17) and their multiples.
In similar triangles, if the sides are in the ratio k, the areas of the triangles are in the ratio k^2.
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Re: In the figure given below, the length of PQ is 12 and the length of PR  [#permalink]

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28 Apr 2017, 19:49
Excellent Question.

Here is what i did on this one ->

In Triangle PQR => Using pythagorus => PQ=12,PR=15 and =>QR=9
Area PQR=1/2 *12*9 => 54

As per question = 54-Area(STU)=Area(STU)
2*Area(STU)=> 54
Area(STU)=27

For the ease of writing=>Let TU=x
ST=y

Area (STU)=> xy/2 = 27
Hence xy=54

Now x/y=12/9
Multiplying the two above equations => y^2=72 => y=6√2

Hence x=> 54/(6√2) => (9√2)/2

SMASH THAT C.

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Re: In the figure given below, the length of PQ is 12 and the length of PR &nbs [#permalink] 28 Apr 2017, 19:49
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