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

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Math Expert
Joined: 02 Sep 2009
Posts: 52906
The height of an equilateral triangle is the side of a smaller equilat  [#permalink]

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07 Sep 2016, 03:44
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10
00:00

Difficulty:

55% (hard)

Question Stats:

66% (02:30) correct 34% (02:14) wrong based on 244 sessions

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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 [ 7.15 KiB | Viewed 8863 times ]

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

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09 Sep 2016, 11:09
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Top Contributor
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Bunuel wrote:

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...

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

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

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

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

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

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

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

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

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##### General Discussion
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Joined: 26 Nov 2012
Posts: 592
The height of an equilateral triangle is the side of a smaller equilat  [#permalink]

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07 Sep 2016, 05:38
1
Bunuel wrote:

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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Joined: 17 Mar 2013
Posts: 6
Re: The height of an equilateral triangle is the side of a smaller equilat  [#permalink]

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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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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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15 Oct 2018, 10:09
Bunuel wrote:

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.)

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

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

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

$$? = 1 - {3 \over 4} = {1 \over 4}$$

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   [#permalink] 15 Oct 2018, 10:09
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