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# If P is the perimeter of an equilateral triangle, which of the followi

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Senior RC Moderator
Joined: 02 Nov 2016
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If P is the perimeter of an equilateral triangle, which of the followi  [#permalink]

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18 Feb 2019, 12:08
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25% (medium)

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78% (01:35) correct 22% (01:17) wrong based on 52 sessions

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If P is the perimeter of an equilateral triangle, which of the following represents the height of the triangle?

A. $$\frac{P}{3}$$

B. $$(P\sqrt{3})/3$$

C. $$\frac{P}{4}$$

D. $$(P \sqrt{3})/6$$

E. $$\frac{P}{6}$$

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Re: If P is the perimeter of an equilateral triangle, which of the followi  [#permalink]

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18 Feb 2019, 18:28
2

Solution

Given:
• P is the perimeter of an equilateral triangle

To find:
• The height of the equilateral triangle

Approach and Working:
As the perimeter of the equilateral triangle is P,
• The length of each side of the triangle = $$\frac{P}{3}$$

We know that, for any equilateral triangle the height is equal to $$\frac{√3}{2}$$ times the length of any side.
• Therefore, the height of the triangle = $$\frac{√3}{2} * \frac{P}{3} = \frac{P√3}{6}$$

Hence, the correct answer is option D.

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Re: If P is the perimeter of an equilateral triangle, which of the followi  [#permalink]

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18 Feb 2019, 15:28
Perimeter =P and the equilateral triangle has 3 equal length therefore each length =1/3p ,since 1/3P+3P+1/3P=P
Equilateral triangle ratio is in the form x:x\/3:2x thus 30:60:90 degrees where (x\/3) the height ,(2x) the hypothenus and x is half the base length basically derive from picking any equilateral triangle of say length 2x units and using Pythagorean’s to find height
Since the hypothenus is 1/3p the base length will be 1/2(1/3p)=1/6p
And height =1/6p*\/3=P\/3/6
Ans is D
Hope it helps

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Re: If P is the perimeter of an equilateral triangle, which of the followi  [#permalink]

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18 Feb 2019, 19:03
If P is the perimeter of an equilateral triangle, which of the following represents the height of the triangle?

A. $$\frac{P}{3}$$

B. $$P\sqrt{3}/3$$

C. $$\frac{P}{4}$$

D. $$P \sqrt{3}/6$$

E. $$\frac{P}{6}$$

Since it is an equilateral triangle and its perimeter is P then its side will be P/3. When we draw a height in an equilateral triangle the half portion of the triangle becomes 30-60-90 triangle. If we draw the height in it, the side opposite to 30 degree angle becomes P/6 since every side is P/3. To find the height we can use Pythagoras Theorem.

Let the height be x,

The equation becomes, x^2 + (P/6)^2 = (P/3)^2

=> x^2 = (P/3)^2 - (P/6)^2

=> x^2 = P^2/9 - P^2/36

=> x^2 = (4P^2 - p^2) / 36

=> x^2 = 3P^2 /36

=> x^2 = P^2/12

=> x = P / √ 12

=> x = P * 2 √ 3 / 12

=> x= P √ 3 / 6

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Re: If P is the perimeter of an equilateral triangle, which of the followi  [#permalink]

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19 Feb 2019, 00:54
If P is the perimeter of an equilateral triangle, which of the following represents the height of the triangle?

A. $$\frac{P}{3}$$

B. $$P\sqrt{3}/3$$

C. $$\frac{P}{4}$$

D. $$P \sqrt{3}/6$$

E. $$\frac{P}{6}$$

each side of the equilateral triangle ; P/3
height of equilateral triangle √3/2 * side = √3/2 * P /3; √3P/6
IMO D
Re: If P is the perimeter of an equilateral triangle, which of the followi   [#permalink] 19 Feb 2019, 00:54
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