In the figure shown, PQ is perpendicular to QR and ST is parallel to Q

Question

https://gmatclub.com/forum/download/file.php?id=40996 In the figure shown, PQ is perpendicular to QR and ST is parallel to QR. The measure of SQR is 45°; the measure of TQR is 30°; and QS 50. What is the perimeter of triangle SQT? (A) 25(2 + \sqrt{6} + \sqrt{2}) (B) 25(2 + \sqrt{8}) (C) 25(2 + 2\sqrt{3}) (D) 50(2 + \sqrt{8}) (E) 50(2 + \sqrt{3} + \sqrt{2}) 2018-04-24_0852.png

EgmatQuantExpert · Accepted Answer

Solution

Given:  
• PQ is perpendicular to QR
• ST is parallel to QR
• Angle SQR = 45°
• Angle TQR = 30°
• QS = 50

To find:  
• The perimeter of the triangle SQT

Approach and Working:   
• From points S and T respectively, we can drop perpendicular to QR. Let us assume that those perpendiculars intersect QR at the points A and B respectively

https://e-gmat.com/blogs/wp-content/uploads/2018/04/2A-e1524560928307.png

• In the given figure, SA = TB and ST = AB
• In triangle SAQ, the angles are 45°, 45°, and 90°. Therefore, the sides will follow the ratio 1:1: \sqrt{2} 
 o Here SQ = 50 which is equivalent to  \sqrt{2}  in the ratio, therefore the other two sides SA and AQ are 25 \sqrt{2}  each. 
• In triangle QTB, the angles are 30°, 60°, and 90° respectively. Therefore, the sides will follow the ratio 1: \sqrt{3} :2.
 o Here TB = 25 \sqrt{2} , so QB = 25 \sqrt{6}  and TQ = 50 \sqrt{2}  
• Now, ST = AB = QB – QA = 25 \sqrt{6}  - 25 \sqrt{2}  
• The perimeter of the triangle SQT = SQ + QT + ST = 50 + 50 \sqrt{2}  + 25 \sqrt{6}  - 25 \sqrt{2}  = 50 + 25 \sqrt{2} + 25 \sqrt{6}  = 25(2 +  \sqrt{6}  +  \sqrt{2} )
Hence, the correct answer is Option A

Answer: A

RohitGoyal1123 · Answer

By Dropping perpendiculars from S and T on QR at A and B respectively.
We have 
Triangle SQA as 45,45,90 Triangle and 
Triangle TQB as 60, 30, 90 Triangle
SA = TB and TB = AB, as ST is parallel to QR

Perimeter of Triangle SQT = QS+ST+TQ

Now, in triangle SQT, since this is 45,45,90 Triangle, SA = QA, and applying  1:1:\sqrt{2} , QA = SA =  \frac{50}{\sqrt{2}} = 25\sqrt{2}

Now, in Triangle TQB, using 30,60,90, and SA = TB; QT =  50\sqrt{2}  and QB =  25\sqrt{6}.

Now AB = QB - QA =  25\sqrt{6}  -  25\sqrt{2}

So, Perimeter of Triangle SQT = QS+ST+TQ =  25(2 + \sqrt{6} + \sqrt{2})

A  option

minatminat · Answer

Is there any way to solve it without knowing properties of triangles like 45,45,90 Triangle have a side ratio of 1:1:2

Or this is something I have to memorise?

Thanks

EgmatQuantExpert · Answer

You don't need to memorize it, although it is very easy to remember and calculate
Consider the following figure:
 https://e-gmat.com/blogs/wp-content/uploads/2018/04/MSP1019213e56073h5676f0b000011283h8cfbe8bf29.gif 
Here angleRPQ and angleRQP both equal to 45 degrees, therefore, RP = RQ = a (let's assume) and anglePRQ is 90

as PRQ is a right-angled triangle, we can apply Pythagorean theorem to write:
 PQ2  =  a2  +  a2  
or  PQ2  = 2 a2 
or PQ =  \sqrt{2} a
therefore, the ratio of the sides RP:RQ:PQ = a:a: \sqrt{2} a = 1:1: \sqrt{2}

RohitGoyal1123 · Answer

Hi minatminat no there is no need to memorize these things, they are just the derivations/extensions from Pythagoras Theorem

el2010 · Answer

This question can be narrowed down to two choices without any calculations and by using logic alone. The question asks what is the perimeter for this triangle. Before writing anything down, I look at the answer choices and see how the answer choices are constructed. In this case, it's in factor form.

What answers can I logically eliminate...I need an answer that has three parts since a perimeter of a triangle requires adding three sides. Eliminate B, C, and D. Left between A and E and we know QS=50. Find the answer choices that includes 50 as one of the legs of the triangle. Answer Choice A.

Hero8888 · Answer

We get triangles with 30-60-90 and 45-45-90. It will help us to calculate other parameters based on known side of 50 cm.
See attached file.
Answer A.

minatminat · Answer

Thanks! I have no problem with understanding the ratio for 45:45:90 triangle but I think I will memorise the ratio for 30:60:90 triangle.

minatminat · Answer

Thank. Actually I tried but still got stuck after working out the length for 25 square root 2. I think maybe I will memories the ratio, I guess it’s easier .

push12345 · Answer

Using figure by RohitGoyal1123
After we get angles ∆QST 15,30 and 135

Even if we don't know ratio we can use

a/sinA=b/sinB. Formula to calculate different sides as all angles are known just sides need to find out

Eg
50/sin30=x/sin15=x/sin(45-30)
But one need to have sin and cos formula on tips to solve this

Same for calculation for other side
Give kudos if it helps

Posted from my mobile device

bumpbot · Answer

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In the figure shown, PQ is perpendicular to QR and ST is parallel to Q

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In the figure shown, PQ is perpendicular to QR and ST is parallel to Q

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