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If the perimeter of ΔACD is 9+3√3, what is the perimeter of equilatera

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If the perimeter of ΔACD is 9+3√3, what is the perimeter of equilatera  [#permalink]

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New post 25 Sep 2016, 09:50
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If the perimeter of ΔACD is 9+3√3, what is the perimeter of equilatera  [#permalink]

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New post 25 Sep 2016, 17:22
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The altitude of an equilateral triangle is \(side*\frac{√3}{2}\).

As perimeter of triangle ACD is \(9+3√3\),

AC + CD + AD = \((side+\frac{side}{2}+side*\frac{√3}{2}) = 9 + 3√3\) or side = 6.

Perimeter of equilateral triangle, ABC is 3*side = 18(OptionC)
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If the perimeter of ΔACD is 9+3√3, what is the perimeter of equilatera  [#permalink]

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New post 26 Sep 2016, 04:30
Bunuel wrote:
Image
If the perimeter of ΔACD is 9+3√3, what is the perimeter of equilateral triangle ΔABC?


We know that the angle bisector also bisects the opposite side in an equilateral triangle. Assume, AC=2a. This means AD=a (AB=2a, and C bisects AB). Also height is given by √3(side)/2 = √3a
Perimeter = 2a+a+√3a=3a+√3a=9+3√3. from this, a=3, so side = 2a=6. Perimeter =18
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Re: If the perimeter of ΔACD is 9+3√3, what is the perimeter of equilatera  [#permalink]

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New post 26 Sep 2016, 05:25
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Bunuel wrote:
Image
If the perimeter of ΔACD is 9+3√3, what is the perimeter of equilateral triangle ΔABC?

A. 9
B. 18−3√3
C. 18
D. 18+3√3
E. 27


Attachment:
T6016.png


Note the \(\sqrt{3}\) term. The perimeter will include the altitude AD which would probably be the \(3*\sqrt{3}\) term.

Altitude of an equilateral triangle \(= \sqrt{3}*Side/2 = 3*\sqrt{3}\)

Then Side is 6. If that is true, then the rest of the perimeter would be 6 + 3 (half of base) = 9 (which is correct)

Perimeter of equilateral triangle ABC = 3*6 = 18
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Re: If the perimeter of ΔACD is 9+3√3, what is the perimeter of equilatera  [#permalink]

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New post 26 Sep 2016, 06:17
Bunuel wrote:
Image
If the perimeter of ΔACD is 9+3√3, what is the perimeter of equilateral triangle ΔABC?

A. 9
B. 18−3√3
C. 18
D. 18+3√3
E. 27


Attachment:
T6016.png


Answer: In triangle ACD, angle ACD = 90; angle ACD = 60 (because equilateral triangle) So, <CAD = 30
Therefore sides AC = 2x CD=x & AD= \sqrt{3}x
2x+x+ 3\sqrt{3}x = 3+\sqrt{3}x
So, x= 3

Hence, AC = 6 Perimeter = 6
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If the perimeter of ΔACD is 9+3√3, what is the perimeter of equilatera  [#permalink]

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New post 27 Jan 2018, 03:21
1
Bunuel wrote:
Image
If the perimeter of ΔACD is 9+3√3, what is the perimeter of equilateral triangle ΔABC?

A. 9
B. 18−3√3
C. 18
D. 18+3√3
E. 27


Attachment:
T6016.png


I solved it in a different way, please let me know if I am taking any risks here:

we know that the side opposite to angle 60 (altitude) = x √3 (1:√3:2)

so if we consider "3√3" in ACD perimeter "9+ 3√3 " is the length of the altitude, then AC+ CD =9

Since this is equilateral triangle then the perimeter = 9*2=18 (C)
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If the perimeter of ΔACD is 9+3√3, what is the perimeter of equilatera  [#permalink]

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New post 29 Jan 2018, 05:36
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NDND wrote:
Bunuel wrote:
Image
If the perimeter of ΔACD is 9+3√3, what is the perimeter of equilateral triangle ΔABC?

A. 9
B. 18−3√3
C. 18
D. 18+3√3
E. 27


Attachment:
T6016.png


I solved it in a different way, please let me know if I am taking any risks here:

we know that the side opposite to angle 60 (altitude) = x √3 (1:√3:2)

so if we consider "3√3" in ACD perimeter "9+ 3√3 " is the length of the altitude, then AC+ CD =9

Since this is equilateral triangle then the perimeter = 9*2=18 (C)


Yes, you can make that assumption but you do need to confirm once that \(CD:AD:AC = 1:\sqrt{3}:2\)
If \(AD = 3\sqrt{3}\), then CD = 3 and AC = 6. They do add up to 9 and give perimeter as \(9 + 3\sqrt{3}\). All matches up.

The reason you can't assume without confirming is what if the side x is \(\sqrt{3}\) and the side 2x is \(2\sqrt{3}\) and the \(\sqrt{3}x\) is then just 3?
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Re: If the perimeter of ΔACD is 9+3√3, what is the perimeter of equilatera  [#permalink]

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New post 29 Jan 2018, 10:26
VeritasPrepKarishma wrote:
NDND wrote:
Bunuel wrote:
Image
If the perimeter of ΔACD is 9+3√3, what is the perimeter of equilateral triangle ΔABC?

A. 9
B. 18−3√3
C. 18
D. 18+3√3
E. 27


Attachment:
T6016.png


I solved it in a different way, please let me know if I am taking any risks here:

we know that the side opposite to angle 60 (altitude) = x √3 (1:√3:2)

so if we consider "3√3" in ACD perimeter "9+ 3√3 " is the length of the altitude, then AC+ CD =9

Since this is equilateral triangle then the perimeter = 9*2=18 (C)


Yes, you can make that assumption but you do need to confirm once that \(CD:AD:AC = 1:\sqrt{3}:2\)
If \(AD = 3\sqrt{3}\), then CD = 3 and AC = 6. They do add up to 9 and give perimeter as \(9 + 3\sqrt{3}\). All matches up.

The reason you can't assume without confirming is what if the side x is \(\sqrt{3}\) and the side 2x is \(2\sqrt{3}\) and the \(\sqrt{3}x\) is then just 3?


Thanks VeritasPrepKarishma

So, is this applicable on this problem, how can we confirm here? do you advise not to use this way in this problem?

Thx
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Re: If the perimeter of ΔACD is 9+3√3, what is the perimeter of equilatera  [#permalink]

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New post 30 Jan 2018, 02:10
1
NDND wrote:

So, is this applicable on this problem, how can we confirm here? do you advise not to use this way in this problem?

Thx


Here is how you can confirm here:

The total perimeter is \(9 + 3\sqrt{3}\).
If \(AD = 3\sqrt{3}\), then CD = 3 and AC = 6 to get the ratio \(1:\sqrt{3}:2\). CD and AC do add up to 9 and give perimeter as \(9 + 3\sqrt{3}\).
All matches up.
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Re: If the perimeter of ΔACD is 9+3√3, what is the perimeter of equilatera  [#permalink]

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New post 31 Jan 2018, 17:06
Bunuel wrote:
Image
If the perimeter of ΔACD is 9+3√3, what is the perimeter of equilateral triangle ΔABC?

A. 9
B. 18−3√3
C. 18
D. 18+3√3
E. 27


Since triangle ABC is an equilateral triangle, angle ACD is 60 degrees. We also see that angle ADC is 90 degrees, so angle DAC is 30 degrees, and thus triangle ACD is a 30-60-90 right triangle, with base CD = x, altitude AD = x√3, and hypotenuse AC = 2x.

Thus:

3x + x√3 = 9 + 3√3

We see that x must be 3.

So hypotenuse AC = side of triangle ABC = 2x = 2(3) = 6, and thus the perimeter of equilateral triangle ABC is 3 x 6 = 18.

Answer: C
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