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# In the figure shown,PQRSTU is a regular polygon with sides of length x

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In the figure shown,PQRSTU is a regular polygon with sides of length x  [#permalink]

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Updated on: 22 Sep 2019, 14:46
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In the figure shown,PQRSTU is a regular polygon with sides of length x. What is the perimeter of triangle PRT in terms of x?

A. ($$x\sqrt{3}$$)/2

B. $$x\sqrt{3}$$

C. (3$$x\sqrt{3}$$)/2

D. 3$$x\sqrt{3}$$

E. 4$$x\sqrt{3}$$

OG Q 2017(Book Question: 145)

Attachment:

1494909854_591a839e50d63.png [ 5.75 KiB | Viewed 23470 times ]

Originally posted by AbdurRakib on 23 Jun 2016, 11:47.
Last edited by MikeScarn on 22 Sep 2019, 14:46, edited 1 time in total.
Fixed Typo
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Re: In the figure shown,PQRSTU is a regular polygon with sides of length x  [#permalink]

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23 Jun 2016, 19:19
21
16
AbdurRakib wrote:

In the figure shown,PQRSTU is a regular polygon with sides of lenght x.What is the perimeter of triangle PRT in terms of x?

A. ($$x\sqrt{3}$$)/2
B. $$x\sqrt{3}$$
C. (3$$x\sqrt{3}$$)/2
D. 3$$x\sqrt{3}$$
E. 4$$x\sqrt{3}$$

OG Q 2017(Book Question: 145)

Hi,

two ways this Q can be done..

(I) Logic....
It can be clearly seen the Triangle is an equilateral triangle ........... Each side joins edges of two adjacent sides of a regular hexagon...
so our ANS should be a multiple of 3........ ONLY D and C are left...
Now D gives side as $$x\sqrt{3}$$ and C as $$x\sqrt{3}/2 \approx{x*1.7/2}$$.....
Now the side of equilateral triangle cannot be less than the side of hexagon.... so C is also out
ans D...

(II) Method...

hexagon has each angle as 120.....
Total sum of angles = (n-2)*180 = 4*180...
each angle = $$\frac{4*180}{6} = 120$$..

take any triangle formed by two sides x and third side as the side of internal triangle..
so it becomes ISOSCELES triangle with equal sides as x and central angle = 120.... so other two angles are (180-120)/2 = 30..
Draw a perpendicular on third side from central angle, it will meet the THIRD side in center and will form a 30-60-90 triangle....
here opposite to $$90^{\circ}$$= x, so opposite $$60^{\circ}$$ = $$x*\sqrt{3}/2$$, which is nothing but HALF of third side..
so third side = $$2*x*\sqrt{3}/2 = x\sqrt{3}$$...
therefore PERIMETER = $$3*x\sqrt{3}$$
D
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Re: In the figure shown,PQRSTU is a regular polygon with sides of length x  [#permalink]

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23 Jun 2016, 13:49
2

The equilateral triangle inside a regular hexagon of side x will have it's side as $$x*\sqrt{3}$$. Therefore perimeter is 3x*$$\sqrt{3}$$
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Re: In the figure shown,PQRSTU is a regular polygon with sides of length x  [#permalink]

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21 Aug 2016, 04:00
2
Hi, can someone please shed some light on why the triangle inside is equilateral? I understood why the 2 sides are equal since all angles and sides are equal, but what is the concept relating to the third side being equal as well?

Additionally, if it is an equilateral triangle, why do we assume the side to be $$\sqrt{3}x$$?

Thank you.
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Re: In the figure shown,PQRSTU is a regular polygon with sides of length x  [#permalink]

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24 Aug 2016, 04:06
Neeraj91 wrote:
Hi, can someone please shed some light on why the triangle inside is equilateral? I understood why the 2 sides are equal since all angles and sides are equal, but what is the concept relating to the third side being equal as well?

Additionally, if it is an equilateral triangle, why do we assume the side to be $$\sqrt{3}x$$?

Thank you.

The triangle is equilateral because the distances between any two alternate vertices of PQRSTU are the same - PR = RT = TP.
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Re: In the figure shown,PQRSTU is a regular polygon with sides of length x  [#permalink]

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24 Aug 2016, 04:15
Bunuel wrote:
Neeraj91 wrote:
Hi, can someone please shed some light on why the triangle inside is equilateral? I understood why the 2 sides are equal since all angles and sides are equal, but what is the concept relating to the third side being equal as well?

Additionally, if it is an equilateral triangle, why do we assume the side to be $$\sqrt{3}x$$?

Thank you.

The triangle is equilateral because the distances between any two alternate vertices of PQRSTU are the same - PR = RT = TP.

Hi Bunuel,

Thanks for the reply. So just to make sure I understood the concept, in any regular polygon of n sides, not only are all the sides equal, the distances between ANY two vertices are equal. Would I be correct in framing it so?
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Re: In the figure shown,PQRSTU is a regular polygon with sides of length x  [#permalink]

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24 Aug 2016, 04:27
1
1
1
Neeraj91 wrote:
Bunuel wrote:
Neeraj91 wrote:
Hi, can someone please shed some light on why the triangle inside is equilateral? I understood why the 2 sides are equal since all angles and sides are equal, but what is the concept relating to the third side being equal as well?

Additionally, if it is an equilateral triangle, why do we assume the side to be $$\sqrt{3}x$$?

Thank you.

The triangle is equilateral because the distances between any two alternate vertices of PQRSTU are the same - PR = RT = TP.

Hi Bunuel,

Thanks for the reply. So just to make sure I understood the concept, in any regular polygon of n sides, not only are all the sides equal, the distances between ANY two vertices are equal. Would I be correct in framing it so?

No. I'm saying that in this particular case the distances between any two alternate vertices of PQRSTU are the same.

It should be very easy to see: triangles PQR, RST and TUP are congruent (their two sides and the angle between them are equal), thus their third sides also must be equal.
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Re: In the figure shown,PQRSTU is a regular polygon with sides of length x  [#permalink]

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24 Aug 2016, 05:23
Thank you. By congruency test, I understood why they have to be equal theoretically. Visually and logically, it seems obvious they are equal, but I was not sure how to make that conclusion without an underlying theory.
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Re: In the figure shown,PQRSTU is a regular polygon with sides of length x  [#permalink]

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25 Dec 2016, 06:23
Consider Triangle PQR
(a) It is an isosceles triangle hence two smaller angles are equal and = (180-120)/2=30 degree.
(b) Third side of the triangle PQR = cosine projections of 'x' = 2*x*cos 30 = (2)X(x)*(Sqrt (3)/2)=(Sqrt 3)x
(c) Same logic holds for other 2 sides, and hence perimeter = 3*(Sqrt 3)x
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Re: In the figure shown,PQRSTU is a regular polygon with sides of length x  [#permalink]

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11 Jan 2017, 10:19
But if we see PQR as an isosceles triangle than the sides are x:x:x(sqrt 2). So thats why I came to the solution 3 times x(sqrt 2) = 3x(sqrt2)
Could somebody please explain where I did a mistake?

EDIT: Sorry, I got it now.
The isosceles triangles needs a right triangle!
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Re: In the figure shown,PQRSTU is a regular polygon with sides of length x  [#permalink]

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05 Apr 2017, 23:46

PQRSTU is a regular polygon with sides of length x. Its area = 6* (√3/4) *x^2 [since a regular polygon is made of 6 equilateral triangles of side x]

If you can visualize, triangle PRT has half the area of the polygon, [joining vertices P,R,T to the centre]

Area of triangle PRT with side "a" = (√3/4)*a ^2 = (1/2) * area of PQRSTU = (1/2) * 6* (√3/4) *x^2 = 3 * x^2

therefore a= √3 * x

perimeter of triangle PRT = 3 *a = 3 *√3 * x

Alternative Note: The equilateral triangle inside a regular hexagon of side x will have it's side as x ∗ √3
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Re: In the figure shown,PQRSTU is a regular polygon with sides of length x  [#permalink]

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09 Oct 2017, 05:15
I am completely missing something on this one:

Once you split the Equilateral triangle into two 30-60-90 triangles, in terms of x the sides should be x-x(sqrt 3)-2x.

The hypotenuse would be: 2x which is one side of the equilateral. So perimeter for the equilateral would 2x+2x+2x.

Obviously I am wrong, but something just isn't sticking with this one for some reason...
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Re: In the figure shown,PQRSTU is a regular polygon with sides of length x  [#permalink]

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09 Oct 2017, 13:09
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1
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AbdurRakib wrote:

In the figure shown,PQRSTU is a regular polygon with sides of lenght x.What is the perimeter of triangle PRT in terms of x?

A. ($$x\sqrt{3}$$)/2
B. $$x\sqrt{3}$$
C. (3$$x\sqrt{3}$$)/2
D. 3$$x\sqrt{3}$$
E. 4$$x\sqrt{3}$$

OG Q 2017(Book Question: 145)

Quote:
Roosterbooster wrote: I am completely missing something on this one

Attachment:

hhhhhhh.png [ 10.32 KiB | Viewed 31625 times ]

Roosterbooster , I think it is easier to draw the right 30-60-90 triangles from the angle of the hexagon. See diagram.

• Dividing the equilateral triangle in the center? Doable. Not easy.

If you divide the triangle into six (not two) congruent 30-60-90 triangles after reading what is below, it will work (but I think drawing from vertex U is easier)

You cannot solve for a side of the triangle very easily by dividing the triangle with one median. Try all three medians. Then your scale is easier to correlate with the side of the hexagon.

I think you also just forgot to divide the ratio by 2. Easy mistake.

Find one side length of the hexagon to find perimeter

Start by dropping an altitude from one vertex to the side of the internal equilateral triangle

What we know:
-- it is a regular hexagon

-- 120° at each vertex
Because this shape is a regular hexagon, each vertex is 120 degrees.
Total interior degrees of a polygon = 180(n-2) where n = number of sides, here, 6
Total interior degrees of this hexagon: (4 * 180) = 720°
Degree measure per vertex angle: 720 degrees/6 angles = 120 degrees per angle

-- The triangle PQR is equilateral
Because the shape is a regular hexagon, the distance between two alternating vertices is equal.
Thus PR = RT = TP. The triangle is equilateral.

• Next step: Notice 120° at each vertex. We are always looking for 30-60-90 or 45-45-90 triangles
∆ PTU is isosceles: PU = UT
Drop an altitude from vertex U.
That altitude is a perpendicular bisector
-- Now there are two 60° angles at vertex U.
-- The altitude is perpendicular. It creates two 90° angles.
-- ∆ PTU 's other two angles each must = 30°
120° + small ∠ + small ∠ = 180°
Two small angles = (180° - 120° = 60° total
Each small angle = 30°

• From the two 30-60-90 triangles we can find the length of the triangle's side.

There are two 30-60-90 right triangles, but their sides have been scaled down

-- a 30-60-90 triangle has sides in ratio $$x: x\sqrt{3}: 2x$$
-- the ratio of sides holds, but each part of the ratio must be divided by 2

Why divide the ratio by 2?

Because the sides of ∆ PTU are defined by the hexagon's side length,
and the hexagon's side length, we are told, $$x$$

The side of the hexagon is the side opposite the triangle's right angle. Work backwards. Label that side $$x$$.

$$x$$ is still opposite the right angle, but it has been scaled down.
The side opposite the 90° is usually $$2x$$

If $$2x$$ has been scaled down to $$x$$, we must divide all else by 2.

[Your way: You can divide the equilateral triangle into six congruent right 30-60-90 triangles, and
The side labeled $$\frac{x\sqrt{3}}{2}$$ will still be opposite a 60-degree angle --]

but we cannot label the side length opposite the 90 degree angle "$$2x$$"
no matter where we draw the right triangles

It is not $$2x$$. Defined by the hexagon, it is $$x$$

• Divide all three parts of the ratio

-- $$\frac{2x}{2} = x$$ - that is the side opposite the right angle

If I divide one part of a ratio by 2, I must divide all terms by 2, so

$$\frac{x}{2}$$ = the side opposite the 30 degree angle
(normally the side opposite the 30° angle is $$x$$)

Finally, $$\frac{x\sqrt{3}}{2}$$ is the side opposite the 60 degree angle
(that side is based on $$x$$; $$x$$ is now $$\frac{x}{2}$$, so $$\frac{x}{2}$$ gets multiplied by $$\sqrt{3}$$)

• Find the side length of the equilateral triangle, then its perimeter

From the diagram: each side of the equilateral triangle has length

$$\frac{x\sqrt{3}}{2}$$ + $$\frac{x\sqrt{3}}{2}$$ = $$x\sqrt{3}$$

The perimeter? $$x\sqrt{3}$$ + $$x\sqrt{3}$$ + $$x\sqrt{3}$$, OR

$$3x\sqrt{3}$$

Hope it helps.
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Re: In the figure shown,PQRSTU is a regular polygon with sides of length x  [#permalink]

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12 Oct 2017, 16:58
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AbdurRakib wrote:

In the figure shown,PQRSTU is a regular polygon with sides of lenght x.What is the perimeter of triangle PRT in terms of x?

A. ($$x\sqrt{3}$$)/2
B. $$x\sqrt{3}$$
C. (3$$x\sqrt{3}$$)/2
D. 3$$x\sqrt{3}$$
E. 4$$x\sqrt{3}$$

Since we have a regular polygon with 6 sides, each vertex has an angle of 180(6 - 2)/6 = 360/6 = 120 degrees.

If we drop a height from S to RT, which we can call SV, we see that we have created two 30-60-90 right triangles. In particular, SRV is one of them, with side SR as the hypotenuse, side SV opposite the 30-degree angle, and side RV opposite the 60-degree angle. Since SR = x, SV = x/2 (the shortest side is ½ of the hypotenuse in a 30-60-90 right triangle) and RV = x√3/2 (the side opposite the 60-degree angle is √3 times the shortest side).

We can also see that RT is twice RV; thus, RT = 2(x√3/2) = x√3. Since triangle PRT is an equilateral triangle and RT = x√3, the perimeter of triangle PRT is 3(x√3) = 3x√3.

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Re: In the figure shown,PQRSTU is a regular polygon with sides of length x  [#permalink]

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18 Oct 2017, 20:18
1
Boom. Got it. Thanks for that write up!

genxer123 wrote:
AbdurRakib wrote:

In the figure shown,PQRSTU is a regular polygon with sides of lenght x.What is the perimeter of triangle PRT in terms of x?

A. ($$x\sqrt{3}$$)/2
B. $$x\sqrt{3}$$
C. (3$$x\sqrt{3}$$)/2
D. 3$$x\sqrt{3}$$
E. 4$$x\sqrt{3}$$

OG Q 2017(Book Question: 145)

Quote:
Roosterbooster wrote: I am completely missing something on this one

Attachment:
hhhhhhh.png

Roosterbooster , I think it is easier to draw the right 30-60-90 triangles from the angle of the hexagon. See diagram.

If you divide the triangle into six (not two) congruent 30-60-90 triangles after reading what is below, it will work (but I think drawing from vertex U is easier)

You cannot solve for a side of the triangle very easily by dividing the triangle with one median. Try all three medians. Then your scale is easier to correlate with the side of the hexagon.

I think you also just forgot to divide the ratio by 2. Easy mistake.

Regular hexagon, 120 degrees at each vertex, equilateral triangle:

Because this shape is a regular hexagon, each vertex is 120 degrees. (4 * 180 = 720, and 720 degrees/6 angles = 120 degrees per angle)

Also because the shape is a regular hexagon, the distance between two alternating vertices is equal. PR = RT = TP. The triangle is equilateral.

If you draw a perpendicular bisector from one vertex to the side of the triangle, as I have at vertex U, you have two 30-60-90 triangles, and you can find the length of the triangle's side.

Bisect a vertex, there are two 30-60-90 right triangles, but sides have been scaled down

The 30 - 60 - 90 triangle side ratio IS $$x: x\sqrt{3}: 2x$$

Each part of the ratio has been scaled down by half; the ratio between and among sides remains the same.

The triangle's sides are defined by the hexagon's side length of $$x$$.

The side of the hexagon is the side opposite the triangle's right angle. Work backwards. Label that side $$x$$.

$$x$$ is still opposite the right angle, but it has been scaled down

If $$2x$$ has been scaled down to $$x$$, you must divide all else by 2.

You can divide the equilateral triangle into six congruent right 30-60-90 triangles, and
The side labeled $$\frac{x\sqrt{3}}{2}$$ will still be opposite a 60-degree angle --

but you cannot label the side length opposite the 90 degree angle "$$2x$$" no matter where you draw the right triangles

It is not $$2x$$. Defined by the hexagon, it is $$x$$

I compensated.

$$\frac{2x}{2} = x$$ - that is the side opposite the right angle

If I divide one part of a ratio by 2, I must divide all terms by 2, so

$$\frac{x}{2}$$ = the side opposite the 30 degree angle

Finally, $$\frac{x\sqrt{3}}{2}$$ is the side opposite the 60 degree angle (if that side is based on $$x$$, $$x$$ is now $$\frac{x}{2}$$, so $$\frac{x}{2}$$ gets multiplied by $$\sqrt{3}$$)

Find the side length of the equilateral triangle, then its perimeter

From the diagram: each side of the equilateral triangle has length

$$\frac{x\sqrt{3}}{2}$$ + $$\frac{x\sqrt{3}}{2}$$ = $$x\sqrt{3}$$

The perimeter? $$x\sqrt{3}$$ + $$x\sqrt{3}$$ + $$x\sqrt{3}$$, OR

$$3x\sqrt{3}$$

Hope it helps.
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Re: In the figure shown,PQRSTU is a regular polygon with sides of length x  [#permalink]

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08 Mar 2018, 06:34
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Bunuel could you please simplify the application of 30-60-90 triangle here? Really confused.
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Re: In the figure shown,PQRSTU is a regular polygon with sides of length x  [#permalink]

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22 Mar 2018, 09:55
generis wrote:
AbdurRakib wrote:

In the figure shown,PQRSTU is a regular polygon with sides of lenght x.What is the perimeter of triangle PRT in terms of x?

A. ($$x\sqrt{3}$$)/2
B. $$x\sqrt{3}$$
C. (3$$x\sqrt{3}$$)/2
D. 3$$x\sqrt{3}$$
E. 4$$x\sqrt{3}$$

OG Q 2017(Book Question: 145)

Quote:
Roosterbooster wrote: I am completely missing something on this one

Attachment:
hhhhhhh.png

Roosterbooster , I think it is easier to draw the right 30-60-90 triangles from the angle of the hexagon. See diagram.

If you divide the triangle into six (not two) congruent 30-60-90 triangles after reading what is below, it will work (but I think drawing from vertex U is easier)

You cannot solve for a side of the triangle very easily by dividing the triangle with one median. Try all three medians. Then your scale is easier to correlate with the side of the hexagon.

I think you also just forgot to divide the ratio by 2. Easy mistake.

Regular hexagon, 120 degrees at each vertex, equilateral triangle:

Because this shape is a regular hexagon, each vertex is 120 degrees. (4 * 180 = 720, and 720 degrees/6 angles = 120 degrees per angle)

Also because the shape is a regular hexagon, the distance between two alternating vertices is equal. PR = RT = TP. The triangle is equilateral.

If you draw a perpendicular bisector from one vertex to the side of the triangle, as I have at vertex U, you have two 30-60-90 triangles, and you can find the length of the triangle's side.

Bisect a vertex, there are two 30-60-90 right triangles, but sides have been scaled down

The 30 - 60 - 90 triangle side ratio IS $$x: x\sqrt{3}: 2x$$

Each part of the ratio has been scaled down by half; the ratio between and among sides remains the same.

The triangle's sides are defined by the hexagon's side length of $$x$$.

The side of the hexagon is the side opposite the triangle's right angle. Work backwards. Label that side $$x$$.

$$x$$ is still opposite the right angle, but it has been scaled down

If $$2x$$ has been scaled down to $$x$$, you must divide all else by 2.

You can divide the equilateral triangle into six congruent right 30-60-90 triangles, and
The side labeled $$\frac{x\sqrt{3}}{2}$$ will still be opposite a 60-degree angle --

but you cannot label the side length opposite the 90 degree angle "$$2x$$" no matter where you draw the right triangles

It is not $$2x$$. Defined by the hexagon, it is $$x$$

I compensated.

$$\frac{2x}{2} = x$$ - that is the side opposite the right angle

If I divide one part of a ratio by 2, I must divide all terms by 2, so

$$\frac{x}{2}$$ = the side opposite the 30 degree angle

Finally, $$\frac{x\sqrt{3}}{2}$$ is the side opposite the 60 degree angle (if that side is based on $$x$$, $$x$$ is now $$\frac{x}{2}$$, so $$\frac{x}{2}$$ gets multiplied by $$\sqrt{3}$$)

Find the side length of the equilateral triangle, then its perimeter

From the diagram: each side of the equilateral triangle has length

$$\frac{x\sqrt{3}}{2}$$ + $$\frac{x\sqrt{3}}{2}$$ = $$x\sqrt{3}$$

The perimeter? $$x\sqrt{3}$$ + $$x\sqrt{3}$$ + $$x\sqrt{3}$$, OR

$$3x\sqrt{3}$$

Hope it helps.

generis hello i have one awesome question one thing i dont get why are you dividing by two ? by drawing a perpendiculat line we create a right triangle 30 60 90, we are not dividing right triangle, we create it no ?

Any idea niks18 ?
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Re: In the figure shown,PQRSTU is a regular polygon with sides of length x  [#permalink]

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24 Mar 2018, 23:41
dave13 wrote:

generis hello i have one awesome question one thing i dont get why are you dividing by two ? by drawing a perpendiculat line we create a right triangle 30 60 90, we are not dividing right triangle, we create it no ?

Any idea niks18 ?

Hi dave13

I did not get what you mean by dividing by 2? What is being divided by 2 and in which context you are referring?

Secondly the explanations by generis are so detailed that there is hardly anything left for anybody to add to it.
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Re: In the figure shown,PQRSTU is a regular polygon with sides of length x  [#permalink]

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25 Mar 2018, 02:07
niks18 wrote:
dave13 wrote:

generis hello i have one awesome question one thing i dont get why are you dividing by two ? by drawing a perpendiculat line we create a right triangle 30 60 90, we are not dividing right triangle, we create it no ?

Any idea niks18 ?

Hi dave13

I did not get what you mean by dividing by 2? What is being divided by 2 and in which context you are referring?

Secondly the explanations by generis are so detailed that there is hardly anything left for anybody to add to it.

Hi niks18, I agree Generis explanation is great the only thing i didnt get is why all three $$x: x\sqrt{3}: 2x$$
sides we divide by 2 ... if we divide an isolessces triangle into two right triangles than we need need to divide only one part x by 2 as for axample it is explained in this video ...https://www.khanacademy.org/math/geomet ... t-triangle

so yeah thats my confusion, anyways thanks for trying to understand me your patience is appreciated
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Re: In the figure shown,PQRSTU is a regular polygon with sides of length x  [#permalink]

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25 Mar 2018, 04:28
1
dave13 wrote:
niks18 wrote:
dave13 wrote:

generis hello i have one awesome question one thing i dont get why are you dividing by two ? by drawing a perpendiculat line we create a right triangle 30 60 90, we are not dividing right triangle, we create it no ?

Any idea niks18 ?

Hi dave13

I did not get what you mean by dividing by 2? What is being divided by 2 and in which context you are referring?

Secondly the explanations by generis are so detailed that there is hardly anything left for anybody to add to it.

Hi niks18, I agree Generis explanation is great the only thing i didnt get is why all three $$x: x\sqrt{3}: 2x$$
sides we divide by 2 ... if we divide an isolessces triangle into two right triangles than we need need to divide only one part x by 2 as for axample it is explained in this video ...https://www.khanacademy.org/math/geomet ... t-triangle

so yeah thats my confusion, anyways thanks for trying to understand me your patience is appreciated

Hi dave13

I got your query now. it's a simple ratio operation. As you mentioned that the triangle is a 30-60-90 triangle, hence its sides will be in ratio $$1:\sqrt{3}:2$$

Now take a look at generis solution. Side opposite to 90 angel is UT and UT=x

so we have $$2=x$$, hence $$1=\frac{x}{2}$$ & $$\sqrt{3}=\frac{x}{2}*\sqrt{3}$$. so you got the lengths of your sides now.
Re: In the figure shown,PQRSTU is a regular polygon with sides of length x   [#permalink] 25 Mar 2018, 04:28

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