Results 1 to 4 of 4

Math Help - Area of equilateral triangle given known distances of a point from the sides.

  1. #1
    Newbie findmehere.genius's Avatar
    Joined
    Apr 2009
    From
    India, UP
    Posts
    18

    Area of equilateral triangle given known distances of a point from the sides.

    yet another question for someone willing to help for I am tired wasting four long pages....
    Q.There is a point inside an equilateral triangle which is at distances
    1, 2 and 3 from the three sides. The area of the triangle is
    A
    not uniquely determinable
    B 6 root3
    C 6
    D 12 root3
    Last edited by mr fantastic; October 28th 2009 at 03:41 AM. Reason: Moved to new thread, title change.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Dec 2007
    From
    Ottawa, Canada
    Posts
    3,069
    Thanks
    65
    Label triangle A,B,C ; P = point inside
    Place D on AB, E on BC, F on AC : PD = 2, PE = 3, PF = 1

    Join PA, PB, PC

    anglePAF = A ; then anglePAD = 60-A
    sin(60-A) / sin(A) = 2

    anglePBD = B ; then anglePBE = 60-B
    sin(60-B) / sin(B) = 3/2

    anglePCF = C ; then anglePCE = 60-C
    sin(60-C) / sin(C) = 3

    Plus if sides of triangle ABC = a, then area = 3a.

    Hope that helps you continue on page 5
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Grandad's Avatar
    Joined
    Dec 2008
    From
    South Coast of England
    Posts
    2,570
    Hello findmehere.genius
    Quote Originally Posted by findmehere.genius View Post
    yet another question for someone willing to help for I am tired wasting four long pages....
    Q.There is a point inside an equilateral triangle which is at distances
    1, 2 and 3 from the three sides. The area of the triangle is
    A
    not uniquely determinable
    B 6 root3
    C 6
    D 12 root3
    See the attached diagram.

    Quadrilateral ANXM is cyclic, since \angle ANX = \angle AMX = 90^o. So \angle NMX = \angle NAX = \theta, say. (Angles in same segment.)

    Similarly \angle NLX = \angle NBX = \phi, say.

    Now in \triangle NMX, NM^2 = 2^2+3^3 - 2.2.3.\cos120^o = 19 (Cosine Rule)

    \Rightarrow NM= \sqrt{19}

    So \frac{\sin\theta}{2}=\frac{\sin120^o}{\sqrt{19}} (Sine Rule)

    \Rightarrow \sin\theta = \sqrt{\frac{3}{19}}

    \Rightarrow \tan\theta = \frac{\sqrt3}{\sqrt{19-3}}=\frac{\sqrt3}{4}

    \Rightarrow AN = \frac{2}{\tan\theta} (from \triangle ANX)

    = \frac{8}{\sqrt3}

    Similarly AQ= \sqrt7, \sin\phi= \sqrt{\frac37}, and \tan\phi = \frac{\sqrt3}{2}

    \Rightarrow NB = \frac{2}{\tan\phi}=\frac{4}{\sqrt3}

    \Rightarrow AB = \frac{8}{\sqrt3}+\frac{4}{\sqrt3}=\frac{12}{\sqrt3  }=4\sqrt3

    So area \triangle ABC = \tfrac12AB.AC\sin60^o=\tfrac12.4\sqrt3.4\sqrt3.\tf  rac{\sqrt3}{2}=12\sqrt3

    Grandad
    Attached Thumbnails Attached Thumbnails Area of equilateral triangle given known distances of a point from the sides.-untitled.jpg  
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    Joined
    Jan 2009
    Posts
    591
    Quote Originally Posted by findmehere.genius View Post
    yet another question for someone willing to help for I am tired wasting four long pages....
    Q.There is a point inside an equilateral triangle which is at distances
    1, 2 and 3 from the three sides. The area of the triangle is

    A
    not uniquely determinable

    B 6 root3

    C 6

    D 12 root3




    In an equilateral triangle,
    if you know the perpendicular distances to all three sides (which is given),
    then that sum will equal the altitude (h) of the triangle (which is 1+2+3 = 6).

    The base (or 1 side of the triangle) is  2 h \tan(30deg)

    <br />
\tan(30deg) = \dfrac{ \sqrt{3} } {3} <br />

    Thus:
     \text{Area} = h^2 \cdot \dfrac{ \sqrt{3}} {3} =  6^2 \cdot \dfrac{ \sqrt{3}} {3} \,\, = \,\, 12 \cdot \sqrt{3}

    as shown previously.

    .
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: December 7th 2010, 01:57 AM
  2. A point within an equilateral triangle
    Posted in the Trigonometry Forum
    Replies: 2
    Last Post: December 7th 2010, 01:48 AM
  3. Area - circles and equilateral triangle.
    Posted in the Geometry Forum
    Replies: 1
    Last Post: October 14th 2010, 03:38 AM
  4. area of an equilateral triangle
    Posted in the Algebra Forum
    Replies: 3
    Last Post: May 31st 2010, 10:02 AM
  5. Replies: 7
    Last Post: July 19th 2008, 06:53 AM

Search Tags


/mathhelpforum @mathhelpforum