Results 1 to 10 of 10
Like Tree5Thanks
  • 1 Post By bjhopper
  • 1 Post By DenisB
  • 2 Post By johng
  • 1 Post By Idea

Thread: Find angle x interior point inside triangle

  1. #1
    Junior Member
    Joined
    Feb 2011
    Posts
    38

    Find angle x interior point inside triangle

    Isosceles triangle ABC (AB=AC) has interior point P such that angle PBC = 10 and PCB = 30. Angle BAC = 80. Find angle APB without using cosine/sine rule.
    Last edited by eskimogenius; Dec 8th 2016 at 05:35 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    Feb 2011
    Posts
    38

    Re: Find angle x interior point inside triangle

    bump.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Feb 2015
    From
    Ottawa Ontario
    Posts
    1,691
    Thanks
    318

    Re: Find angle x interior point inside triangle

    You DO know that we don't do homework, right?

    How far did you get with this problem?
    Can you calculate angles ABP and ACP?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Feb 2011
    Posts
    38

    Re: Find angle x interior point inside triangle

    I apologise, this isn't homework. I'm studying for an exam and this question came up and I needed some help solving it.

    ABP and ACP are relatively simple to solve using the isosceles triangle, ABP = 40, ACP = 20.

    I've tried drawing altitudes from P to AB and AC, constructing a cyclic quad, and all sorts of things such as constructing a perpendicular line from A to BC. None of which have enabled me to calculate the angle APB.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member
    Joined
    Nov 2007
    From
    Trumbull Ct
    Posts
    985
    Thanks
    38

    Re: Find angle x interior point inside triangle

    Problem can be solved by coordinate geometry.Is this covered in your course.
    Thanks from eskimogenius
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Joined
    Feb 2015
    From
    Ottawa Ontario
    Posts
    1,691
    Thanks
    318

    Re: Find angle x interior point inside triangle

    HINT:
    show that BP = AB

    Then you have isosceles triangle ABP with angle ABP = 40
    Thanks from eskimogenius
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor
    Joined
    Dec 2012
    From
    Athens, OH, USA
    Posts
    1,118
    Thanks
    467

    Re: Find angle x interior point inside triangle

    Denis,

    I carefully drew a diagram a week ago and observed that triangle ABP is indeed isosceles. I've been trying to prove this, but can't do it. I give up; could you post your solution?
    Thanks from Plato and eskimogenius
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Super Member
    Joined
    Jun 2013
    From
    Lebanon
    Posts
    742
    Thanks
    341

    Re: Find angle x interior point inside triangle

    BP meets AC at D. Bisector of angle BAC meets BD at O
    O is the circumcenter of ABC, angle OCB=10
    Let OC=OB=OA=R, AB=AC=b, AD=c, CD=b-c, and BD=d
    We want to show BP=b
    Similar triangles OAD, ABD gives
    \frac{R}{b}=\frac{d-R}{c}=\frac{c}{d}=\frac{R+(d-R)}{b+c}=\frac{d}{b+c}
    From this we get
    \frac{c}{d}=\frac{d}{b+c}

    R=\frac{b c}{d}=\frac{b d}{b+c}

    \text{OD}=d-R=\frac{d^2-b c}{d}=\frac{c^2}{d}

    CP bisects angle OCD, angle OCP = angle PCD = 20 = angle COP
    Triangle OPC is isosceles and CP=OP

    Similar triangles OCD, CPD

    \frac{\text{OC}}{\text{CP}}=\frac{\text{OD}}{\text  {CD}}=\frac{R}{\text{OP}}=\frac{\text{OD}}{b-c}

    From this we get \text{OP}=\frac{b(b-c)}{c}

    and

    \text{BP}=R+\text{OP}=\frac{b d}{b+c}+\frac{b(b-c)}{c}=\frac{b\left(c d+b^2-c^2\right)}{c(b+c)}=b

    For the last equality note that b^2=c^2+d^2-c d in triangle ABD using the fact that angle ADB = 60 so we don't have to say 'law of cosines'
    Last edited by Idea; Dec 18th 2016 at 12:45 AM.
    Thanks from eskimogenius
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Junior Member
    Joined
    Feb 2011
    Posts
    38

    Re: Find angle x interior point inside triangle

    Thank you so much for the solution.

    I have one question, how did you determine/prove that circumcenter O lies on BD?
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Super Member
    Joined
    Jun 2013
    From
    Lebanon
    Posts
    742
    Thanks
    341

    Re: Find angle x interior point inside triangle

    Quote Originally Posted by eskimogenius View Post
    Thank you so much for the solution.

    I have one question, how did you determine/prove that circumcenter O lies on BD?
    OAB iso since angle OBA = angle OAB = 40 so OA=OB

    OB=OC since O is on the perpendicular bisector of side BC

    OA=OB=OC
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: Jul 11th 2015, 05:55 AM
  2. Replies: 1
    Last Post: Oct 10th 2012, 01:18 AM
  3. Triangle inside of circle on graph - angle problem
    Posted in the Trigonometry Forum
    Replies: 4
    Last Post: Jan 25th 2011, 01:58 AM
  4. Replies: 2
    Last Post: Dec 7th 2010, 02:57 AM
  5. Replies: 1
    Last Post: Jun 18th 2008, 06:17 AM

Search tags for this page


/mathhelpforum @mathhelpforum