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An equilateral triangle has a height of

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An equilateral triangle has a height of  [#permalink]

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New post 16 Sep 2018, 07:25
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An equilateral triangle has a height of 4\(\sqrt{3}\). What is the area of this triangle?

A) 16
B) \(16√3\)
C) \(24√3\)
D) 48
E) \(32√3\)

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Re: An equilateral triangle has a height of  [#permalink]

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New post 16 Sep 2018, 08:04
Harshgmat wrote:
An equilateral triangle has a height of 4\(\sqrt{3}\). What is the area of this triangle?

A) 16
B) \(16√3\)
C) \(24√3\)
D) 48
E) \(32√3\)


let each side be a
the height will divide the triangle into two
each of the two will be a right triangle
using Pythagoras , we have
16*3 = \(a^2\) - (\(a^2\)/4)
upon solving we get a = 8
area = \(\frac{1}{2}\) 8 * 4\(\sqrt{3}\) = \(16√3\)
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Re: An equilateral triangle has a height of  [#permalink]

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New post 16 Sep 2018, 16:52
Harshgmat wrote:
An equilateral triangle has a height of 4\(\sqrt{3}\). What is the area of this triangle?

A) 16
B) \(16√3\)
C) \(24√3\)
D) 48
E) \(32√3\)

Attachment:
equitriHfindA 09.16.18.jpg
equitriHfindA 09.16.18.jpg [ 37.97 KiB | Viewed 339 times ]

Good question +1 to refresh a handy tool: the altitude of an equilateral triangle divides it into two congruent 30-60-90 triangles

I. 30-60-90 triangles[/u]
This method is as quick as the formula below and IMO easier to remember.
In an equilateral triangle, each altitude = height = median

Drop an altitude, BD, from ∠B to side AC

The height, BD, divides ∆ABC into two congruent 30-60-90 triangles.
• Altitude BD is a perpendicular bisector of vertex B and opposite side AC

Resultant angles are 30, 60, and 90 degrees
• Original angles A, B, and C = 60°
• ∠ B is bisected into two 30° angles
• At D, the altitude forms two 90° angles
• Angles A and C = 60°
• Both smaller triangles are 30-60-90

30-60-90 triangles have corresponding sides opposite those angles in the ratio
\(x:x\sqrt{3}:2x\)

Height of \(4\sqrt{3}\), opposite the 60° angle, corresponds with \(x\sqrt{3}\)
\(4\sqrt{3}=x\sqrt{3}\)
\(x=4\)
Base AC = BC = \(2x=8\)
(In ∆BCD, side BC is opposite the 90° angle. Sides are equal.)

Area, \(A=\frac{b*h}{2}=\frac{8*4\sqrt{3}}{2}=16\sqrt{3}\)

Answer B

II. Formula
For an equilateral triangle with side \(a\),
Height = \(\frac{1}{2}a*\sqrt{3}\)
Given: height = \(4\sqrt{3}\)
Side length:
\(4\sqrt{3}=\frac{1}{2}a*\sqrt{3}\)
\(8\sqrt{3}=a\sqrt{3}\)
\(a=8\) = base (all sides are equal)

Area, \(A=\frac{b*h}{2}=\frac{8*4\sqrt{3}}{2}=16\sqrt{3}\)

Answer B
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Re: An equilateral triangle has a height of   [#permalink] 16 Sep 2018, 16:52
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