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

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Difficulty:   35% (medium)

Question Stats: 75% (01:52) correct 25% (02:12) wrong based on 126 sessions

### HideShow timer Statistics  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 [ 4.49 KiB | Viewed 3504 times ]

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

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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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Bunuel wrote: 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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Bunuel wrote: 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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GMAT 1: 600 Q40 V31 Re: If the perimeter of ΔACD is 9+3√3, what is the perimeter of equilatera  [#permalink]

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Bunuel wrote: 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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Bunuel wrote: 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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NDND wrote:
Bunuel wrote: 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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Karishma
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Status: Enjoying the Journey
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Joined: 26 Sep 2017
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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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VeritasPrepKarishma wrote:
NDND wrote:
Bunuel wrote: 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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"it takes more time to fix a mistake than to avoid one"
"Giving kudos" is a decent way to say "Thanks" and motivate contributors. Please use them, it won't cost you anything

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GMAT Club Live: 5 Principles for Fast Math: https://gmatclub.com/forum/gmat-club-live-5-principles-for-fast-math-251028.html#p1940045
The Best SC strategies - Amazing 4 videos by Veritas: https://gmatclub.com/forum/the-best-sc-strategies-amazing-4-videos-by-veritas-250377.html#p1934575
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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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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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Bunuel wrote: 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.

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