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505-555 (Easy)|   Geometry|      
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siddhans
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In the figure shown, the length of line segment QS is Updated on: 14 Oct 2019, 07:56
Originally posted by siddhans on 23 Jul 2011, 20:47.
Last edited by Bunuel on 14 Oct 2019, 07:56, edited 4 times in total.
Edited the question and added the OA
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C
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15% (low)

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78% (01:12) correct 22% (01:46) wrong based on 460 sessions

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In the figure shown, the length of line segment QS is \(4\sqrt{3}\). What is the perimeter of equilateral triangle PQR?

A. \(12\)
B. \(12 \sqrt{3}\)
C. \(24\)
D. \(24 \sqrt{3}\)
E. \(48\)

Show SpoilerWhat am i doing wrong here ?
Attachment:
Triangle.png
Triangle.png [ 9.81 KiB | Viewed 24681 times ]
What am i doing wrong here ?

PQ^2 = QS^2 + PS^2

PQ^2 = PS^2 + 48 ------ (1)

QR^2 = SR^2 + 48 ------(2)

Adding 1 and 2


PQ^2 + QR^2 = PS^2 + SR ^2 + 96 ----- 3

But, PS^2 + SR ^2 = PR ^2 ------- 4

Substitute 4 in 3....

PQ^2 + QR^2 = PR^2 + 96 -----5

PQ = PR = QR since equilateral triangle

Therefore PQ^2 = 96


PQ = sqrt (96)


But this doesnt yield the correct answer ....
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C
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BrentGMATPrepNow
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In the figure shown, the length of line segment QS is 19 Dec 2018, 09:39
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siddhans
Attachment:
Triangle.png
In the figure shown, the length of line segment QS is \(4\sqrt{3}\). What is the perimeter of equilateral triangle PQR?

A. \(12\)
B. \(12 \sqrt{3}\)
C. \(24\)
D. \(24 \sqrt{3}\)
E. \(48\)
Show more

Let's first add the given information to the diagram.
If ∆PQR is an equilateral triangle, then the 3 angles are each 60°


At this point, we can see we have a special 30-60-90 right triangle, which we can compare with the BASE 30-60-90 right triangle


When we compare the sides that are opposite the 60° angle, we can see that 4√3 is 4 times the value of √3
So, the magnification factor is 4.
This means the other 2 lengths will be 4 times the sides of the BASE 30-60-90 right triangle


Now that we know that each side has length 8, the perimeter = 8 + 8 + 8 = 24

Answer: C

Cheers,
Brent
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In the figure shown, the length of line segment QS is 23 Jul 2011, 21:08
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siddhans
https://img210.imageshack.us/img210/3589/gmatprepr.jpg



Show SpoilerWhat am i doing wrong here ?
What am i doing wrong here ?




PQ^2 = QS^2 + PS^2

PQ^2 = PS^2 + 48 ------ (1)

QR^2 = SR^2 + 48 ------(2)

Adding 1 and 2


PQ^2 + QR^2 = PS^2 + SR ^2 + 96 ----- 3

But, PS^2 + SR ^2 = PR ^2 ------- 4

Substitute 4 in 3....

PQ^2 + QR^2 = PR^2 + 96 -----5

PQ = PR = QR since equilateral triangle

Therefore PQ^2 = 96


PQ = sqrt (96)


But this doesnt yield the correct answer ....
Show more

Red bold is not true... PS+SR = PR... so if you square (PS+SR)^2 = PR2... not what you have written

simplest way to solve this problem is sin60 = \(\sqrt{3}\)/2 = QS/PQ = 4\(\sqrt{3}\)/PQ
==> PQ = 8
so perimeter = 24
choice C.
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In the figure shown, the length of line segment QS is 21 Mar 2012, 04:07
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agdimple333 is simple and easy. But how do I solve it without knowing sin (60)?
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In the figure shown, the length of line segment QS is 21 Mar 2012, 04:34
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agdimple333 is simple and easy. But how do I solve it without knowing sin (60)?
Show more

Trigonometry is not tested on the GMAT, which means that EVERY GMAT geometry question can be solved without it.

In the figure shown, the length of line segment QS is \(4\sqrt{3}\). What is the perimeter of equilateral triangle PQR?
A. \(12\)
B. \(12 \sqrt{3}\)
C. \(24\)
D. \(24 \sqrt{3}\)
E. \(48\)
Attachment:
Triangle.png
Triangle.png [ 9.81 KiB | Viewed 21516 times ]

Since triangle PQR is equilateral then its all angles equal to 60°. So, right triangle QSR is a 30°-60°-90° right triangle (with angle R equal to 60°). Next, in a right triangle where the angles are 30°, 60°, and 90° the sides are always in the ratio \(1 : \sqrt{3}: 2\) and the leg opposite 60° (QS) corresponds with \(\sqrt{3}\), which makes QR equal to \(4\sqrt{3}*\frac{2}{\sqrt{3}}=8\) (\(\frac{QS}{QR}=\frac{\sqrt{3}}{2}\) --> \(QR=QS*\frac{2}{\sqrt{3}}=8\)).

Thus the perimeter is 3*8=24.

Answer: C.

For more on this subject check Triangles chapter of Math Book: math-triangles-87197.html

Hope it helps.
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In the figure shown, the length of line segment QS is 10 Feb 2016, 09:35
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Here is how I solved it.

An equilateral triangle has 3 equal sides, thus each angle is 60 degrees. Equilateral PQR has a line right down the middle, dividing the triangle PQR into two separate 30-60-90 triangles.



The properties of the 30-60-90 triangles are as follows:
Side with an angle of 30 degrees has a side length of \(x\)
Side with an angle of 60 degrees has a side length of \(x\sqrt{3}\)
Side with an angle of 90 degrees has a side length of \(2x\)

Knowing that \(4\sqrt{3}\) = \(x\sqrt{3}\) -----> \(4 = x\) (the side length of the 60 degree angle), you can easily calculate the rest of the other sides of the triangle.
So, putting it all together the perimeter of Equilateral PQR (see attached figure) = \(8 + 8 + 4 + 4 = 24\)
Attachments

Capturegmat.JPG
Capturegmat.JPG [ 19.37 KiB | Viewed 15155 times ]

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In the figure shown, the length of line segment QS is 19 Apr 2016, 06:31
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Height in a an equilateral triangle equals \(\sqrt{3}\)/2 *a, where a is a side of such triangle. Hence a = \(\sqrt{3}\)/2 * a = 4\(\sqrt{3}\)
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In the figure shown, the length of line segment QS is 19 Apr 2016, 08:30
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siddhans
Attachment:
Triangle.png
In the figure shown, the length of line segment QS is \(4\sqrt{3}\). What is the perimeter of equilateral triangle PQR?

A. \(12\)
B. \(12 \sqrt{3}\)
C. \(24\)
D. \(24 \sqrt{3}\)
E. \(48\)

Show more

Altitude of a equilateral triangle is \({\sqrt{3} * side}/2\)

\({\sqrt{3} * side}/2\) = \(4\sqrt{3}\).

So sides are 8

Perimeter of an equilateral triangle is 3* sides = 3*8 =>24

Hence answer is (C)
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In the figure shown, the length of line segment QS is 15 Aug 2023, 12:39
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