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The height of an equilateral triangle is the side of a smaller equilat

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The height of an equilateral triangle is the side of a smaller equilat  [#permalink]

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New post 07 Sep 2016, 03:44
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The height of an equilateral triangle is the side of a smaller equilateral triangle, as shown above. If the side of the large equilateral triangle is 1, what is AB?

A. 1 - √3/2
B. 0.25
C. 2 - √3
D. 1/3
E. 1 - √3/4

Attachment:
T6036.png
T6036.png [ 7.15 KiB | Viewed 8249 times ]

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Re: The height of an equilateral triangle is the side of a smaller equilat  [#permalink]

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New post 09 Sep 2016, 11:09
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Bunuel wrote:
Image
The height of an equilateral triangle is the side of a smaller equilateral triangle, as shown above. If the side of the large equilateral triangle is 1, what is AB?

A. 1 - √3/2
B. 0.25
C. 2 - √3
D. 1/3
E. 1 - √3/4

Attachment:
T6036.png


The length of one side of the large equilateral triangle is 1...
Image

The line representing the height of that triangle will cut that side in half, which means the one side (indicated below) will have length 1/2
Image

Next, since the smaller triangle (in blue below) is also an equilateral triangle, each angle in that triangle is 60 degrees.
Image

This means the angle adjacent to the 60 degree angle must be 30 degrees....
Image

Also, since the bigger triangle (in red below) is an equilateral triangle, each angle in that triangle is 60 degrees.
Image

Now let's focus our attention on the small red triangle (below)
Image

Since we know 2 of the angles in the triangle, we can see that the 3rd angle is 90 degrees.
Image

This small red triangle is a SPECIAL 30-60-90 triangle, and when we compare it to our "base" 30-60-90 triangle (in blue below), we see that the side opposite the 90-degree angle has length 2 in the base triangle, and the same corresponding side in the red triangle has length 1/2
Image

This tells us that the red triangle is 1/4 the size of the blue base triangle.
So, since the side opposite the 30-degree angle has length 1 in the base triangle, and the length of the corresponding side in the red triangle = 1(1/4) = 1/4
Image

Answer:

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The height of an equilateral triangle is the side of a smaller equilat  [#permalink]

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New post 07 Sep 2016, 05:38
1
Bunuel wrote:
Image
The height of an equilateral triangle is the side of a smaller equilateral triangle, as shown above. If the side of the large equilateral triangle is 1, what is AB?

A. 1 - √3/2
B. 0.25
C. 2 - √3
D. 1/3
E. 1 - √3/4

Attachment:
T6036.png


Height of equilateral triangle is √3/2(a)

Given side of larger triangle is a = 1 then height of larger triangle then height is √3/2(1) = √3/2 and this is side of smaller equilateral triangle.

We have side of smaller triangle and height of smaller triangle is √3/2(a) => √3/2(√3/2) = 3/4.

We have height of smaller triangle.

AB is on larger equilateral triangle and we have side of larger triangle and height of smaller triangle to get AB

AB = 1-3/4 = 1/4 = 0.25.

IMO option B.
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Re: The height of an equilateral triangle is the side of a smaller equilat  [#permalink]

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New post 07 Sep 2016, 19:14
2
the height of the equilateral triangle is also perpendicular bisector for the larger equilateral triangle. So it cuts the side into half. so its 0.5. smaller triangle is equilateral so one of the angle is 60. the other angle would be 180-(90+60) = 30. smaller triangle ABX is 30-60-90 triangle so the side AB will be 0.25.
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Re: The height of an equilateral triangle is the side of a smaller equilat  [#permalink]

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New post 07 Sep 2016, 19:42
If the length of the side of equilateral triangle is a, its height will be \(( \sqrt{3} / 2 ) * a\)

So height of bigger triangle using above formula = \(( \sqrt{3} / 2 ) * 1 = ( \sqrt{3} / 2 )\)

The height of bigger triangle = Length of side of smaller triangle = \(( \sqrt{3} / 2 )\)

Height of Smaller triangle = \(( \sqrt{3} / 2 ) * ( \sqrt{3} / 2 ) = 3/4 = 0.75\)

AB = Length of bigger triangle - Height of slammer triangle = 1 -0.75 = 0.25 (B).
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Re: The height of an equilateral triangle is the side of a smaller equilat  [#permalink]

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New post 15 Oct 2018, 10:09
Bunuel wrote:
Image
The height of an equilateral triangle is the side of a smaller equilateral triangle, as shown above. If the side of the large equilateral triangle is 1, what is AB?

A. 1 - √3/2
B. 0.25
C. 2 - √3
D. 1/3
E. 1 - √3/4

(The formula below we suggest you memorize. It will be used here twice.)

\({h_{eq}} = {{L\sqrt 3 } \over 2}\,\,\,\,\,\left( * \right)\,\,\,\,\,\,\,\,\left( {{\rm{height}}\,\,{\rm{of}}\,\,{\rm{an}}\,\,{\rm{equilateral}}\,\,{\rm{triangle}}\,\,{\rm{with}}\,\,{\rm{side}}\,\,L} \right)\)

Image

\(? = AB = {L_{\,{\rm{large}}}} - {h_{{\rm{eq}}\,{\rm{small}}}}\,\, = \,\,\,1 - \,\,{?_{{\rm{temporary}}}}\)

\(\Delta \,{\rm{small}}\,\,:\,\,\,\,\,{L_{\,{\rm{small}}}} = {h_{\,{\rm{eq}}\,{\rm{large}}}}\,\,\mathop = \limits^{\left( * \right)} \,\,\,\,{{1 \cdot \sqrt 3 } \over 2}\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,{?_{{\rm{temporary}}}} = {h_{{\rm{eq}}\,{\rm{small}}}}\,\,\,\mathop = \limits^{\left( * \right)} \,\,\,{{\sqrt 3 } \over 2} \cdot {{\sqrt 3 } \over 2} = {3 \over 4}\)

\(? = 1 - {3 \over 4} = {1 \over 4}\)

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Re: The height of an equilateral triangle is the side of a smaller equilat &nbs [#permalink] 15 Oct 2018, 10:09
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