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

Question

https://gmatclub.com/forum/download/file.php?id=26379 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 Ques3.jpg

Sagar5999 · Answer

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

AmoyV · Answer

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}}

Answer: C

shriramvelamuri · Answer

yes, Plus one for C. Straight forwrad question.

SherLocked2018 · Answer

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.

sudh · Answer

Given PQ = 12, PR = 15, area of right triangle STU is equal to the area of the shaded region
then Area of  	riangle STU\,=\,\frac{1}{2}  Area of  	riangle PQR

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

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

Answer C

KS15 · Answer

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
Answer C

Bunuel · Answer

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.

Answer (C)

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.

stonecold · Answer

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
Shaded area => 54- Area(STU)

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

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