What is the perimeter of triangle PQR ?

We are looking for the perimeter, so the size of the triangle matters.

St (1) We have all the info about angles and hence the shape ( angles ) of the triangles are fixed but the size is not as the side lengths are not given. Hence insufficient

St (2) Only one altitude is given, but no other info about sides or angles is present. Hence Insufficient!

St (1) + St (2) together : Sufficient! as we have the shape ( all the angles ) as well as the size ( one altitude) of the triangle we can determine the triangle uniquely!

Note, we do not need to find the perimeter, just that a unique triangle is defined is sufficient.
Hence Option (C) is our choice!

Best,
Gladi

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hi Gladiator59

i would like to see detailed solution if you dont mind

thank you

Hey dave13,

Try to imagine this - You have a right-angled triangle but you have no idea how big it is. It could be a minute 3,4,5 unit length sides or it could have side lengths of 3000, 4000 and 5000. So the first triangle has 1000 times smaller side lengths than the second triangle. Each of those has the same angles though.

Obviously, the perimeter will also be a thousand times smaller.

Now, imagine you have a gauge, which rotates (something like a regulator) and you can rotate it to increase the size from 3-4-5 to 3000-4000-5000 - each of the ensuing triangles will have a different perimeter based on how much you rotate the regulator.

Now, when the altitude of the triangle is given - You can no longer rotate the regulator as it becomes fixed at a point ( to agree with the altitude) at this point the triangle is fixed and hence it's perimeter can be found.

Hope this makes things clear.

Best,
Gladi
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Given: A triangle PQR. But there is no information on the type of triangle.
Needed: Type of triangle and depending on the type, the value of sides. For instance, if the triangle is equilateral, then length of one side or any relevant information to find the side length is enough or if the triangle is right triangle we will need the measure of one side and one of the other 2 angles etc.

STATEMENT 1:
Measure of each angle is given, From this we can infer that the triangle is 30-60-90 right triangle.
x+2x+3x=180
so, x=30, 2x=60, 3x=90
But to find the perimeter measure of any one side is needed.
So, this statement is not sufficient.

STATEMENT 2:
From this we know, PQR can be divided into 2 right triangles and the value of height.
But we do not know if the 2 triangles are 45-45-90 or 30-60-90 or a mix of the two or any other angle for that matter. Here, basically we need information about the angles of the 2 right triangles, so that length of sides can be found.
So, we cannot deduce the length of each side of PQR. Hence, this statement is insufficient.

COMBINE:
We know PQR is a right triangle and it is divided into 2 right triangles.
Perpendicular from Q to PR will bisect ∠PQR . Therefore, all the angles for the 2 smaller right triangles can be found.
Clearly, the measure of each side can now be calculated using angles and using the length of perpendicular from Q (maybe using trigonometry, but the actual measure is not needed for DS question) and hence, the perimeter of triangle PQR.
OPTION C

Gladiator59 thank you

Okay second try

from both statements we know that we have right angle \(x\), \(2x\), \(x\sqrt{3}\) and its height is \(4\)

In other words \(x\sqrt{3}\) = \(4\sqrt{3}\)

divide both sides by \(\sqrt{3}\) and get \(x =4\)

so the shortest leg is 4
the hypotenuse is 8
and height \(4\sqrt{3}\)

so perimetre of of right triangle is 4+8+\(4\sqrt{3}\) = 12+\(4\sqrt{3}\)


perimetre of right triangle is 12+\(4\sqrt{3}\)


is it correct answer ?

Looks like it is a bit off dave13. I got perimeter as \(4(1+\sqrt{3})\)

Please find my working in the image attached ( I was bored to type it all! )

Let me know if it makes sense.

Best,
Gladi



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Hi Gladi, Gladiator59, I hope my additional questions wont make you angry


many thanks for taking time to write the detailed solution, but i still have questions:


1. if Altitude is from Q to PR then why you put 4 as the base or may be you drew this triangle it upside down ?


2. I dont understand why x which is hypotenuse equals \(\frac{8}{\sqrt{3}}\) if triangle is in the ratio x, 2x, 3x

where \(x\) is shortest leg, \(2x\) is a hypotenuse, and \(x\sqrt{3}\) then X a hypotenuse is 2*4 =8 can you please explain. based on which rule you make such division?


3. Same question for y, why you write \(y=4/\sqrt{3}\) shouldnt y be equal \(4\sqrt{3}\) based on which rule you make such division?

4. All in all if triangle is 30 60 90 and shortest leg is 4 which an altitude. then logically \(x = 4\), \(2x = 8\) and \(x\sqrt{3} = 4\sqrt{3}\) this is how I think, why ? what`s wrong with my reasoning?


hello pushpitkc chetan2u maybe you can help ?

These are questions I have so far

have a great weekend

Hi chetan2u, Bunuel, I have a quick question on this. I understand the logic that when altitude is given for a right angled triangle we can find all other sides. But can this logic be safely applied to all types of triangles - acute, obtuse i.e. by just knowing all angles and altitude from one side to base, can we determine all sides?

I assume we can by applying trigonometry.

Yes, if you have all angles and one side of a triangle, you can draw a unique triangle.
This means there can be only one type of triangle possible. If you have a unique triangle, you can measure all sides by trigonometry, or draw and measure.
In DS, you will not be required to measure, simply knowing that it will be unique you can answer.
In PS, you will see only a right angled triangle or similar triangles if you have to measure a side.
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can someone answer my questions in my post above ? or share some link OR both please be kind

So, you are clear that it is 30-60-90 triangle ..
(1) The measures of \(\angle\)PQR, \(\angle\)QRP, and \(\angle\) RPQ are \(x^{\circ}\), \(2x^{\circ}\), and \(3x^{\circ}\), respectively.
\(\angle\)PQR=30, \(\angle\)QRP=60, and \(\angle\) RPQ=90

(2) The altitude of \(\triangle\) PQR from Q to PR is 4.
so Q to P is 4, as vertex P is 90..

combined
so sides opposite \(\angle\)PQR=30, \(\angle\)QRP=60, and \(\angle\) RPQ=90 are PR, QP, RQ in ratio 1:\(\sqrt{3}\);2
QP is 4 and is \(\sqrt{3}x\), so x = \(4/\sqrt{3}\) and 2x is \(4/\sqrt{3}\)
Area can be found now
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chetan2u many thanks for explanation BUT one thing i don`t understand is division by \(\sqrt{3}\) Why are we dividing in both cases by \(\sqrt{3}\) based on which rule do you divide by \(\sqrt{3}\) Brent maybe you can help to answer my question GMATPrepNow

All in all if triangle is 30 60 90 and shortest leg is 4 which an altitude. then logically \(x = 4\), \(2x = 8\) and \(x\sqrt{3} = 4\sqrt{3}\) this is how I think, why ? what`s wrong with my reasoning?

The shortest leg is NOT 4....this is where you are going wrong.
ratio of sides are x:\(\sqrt{3}\)x:2x.. the sides are (opposite to 30):(opposite to 60):(opposite to 90)
Now when we check the angles, opposite to 60 angle is 4, so \(\sqrt{3}\)x=4
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chetan2u thank you! finally got it

(1) Only tells us that the triangle is 30-60-90 triangle. But we don't know how big or small it is. Insufficient in determining the perimeter.

(2) Knowing just the altitude gives us no insights into how big the base is or what is the inclination/length of the sides. Insufficient.

Combining the two we get:
30-60-90 triangle with one side as 4
Clearly sufficient in determining the other 2 sides and hence the perimeter

Hence, C.

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