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The Question

My friend recently gave me a problem that I was interested in, but could not completely solve. I've already constructed a diagram that roughly represents the problem, and it's below.

Consider a trapezoid $ABCD$ with $AB \parallel CD$. Additionally, side $BC = CD = 43$ and $AD \perp BD$. Given that the length between the midpoint of diagonal $BD$ and intersectional of diagonals $AC$ and $BD$ is $11$, find the side length $AD$. Express your answer in simplified radical form.

My Understanding

So my first idea was to draw it out, and I've managed to get a good approximation of it. enter image description here

So I gave some names to the other points, calling the midpoint of the diagonal $M$ and the intersection $I$. The first thing that I noted was that because $BC = CD$, I found that drawing it down to the midpoint of diagonal $BD$ created two right triangles. Additionally, I noted congruent angles within the triangle $CPO$ and triangle $ADO$, so I deduced they were similar. But I'm not sure how to proceed from here, as I'm not able to establish the side lengths of any more segments. Does anyone know how to proceed?

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  • $\begingroup$ Why not tell your "friend" you are unable to prove it, and suggest your "friend" create an account and post his/her question here? $\endgroup$
    – amWhy
    Feb 4 at 22:17
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    $\begingroup$ Well, I'm personally interested in finding out more about the problem, so I decided to post it here. I've already spent a bit of time and want to see a solution myself. $\endgroup$ Feb 4 at 22:17
  • $\begingroup$ So then it's your own question. I see not need to cite a "friend", because you it has become your question, right? $\endgroup$
    – amWhy
    Feb 4 at 22:19
  • $\begingroup$ So what's the given length of $MI$? Without it, there is insufficient information. $\endgroup$
    – cosmo5
    Feb 4 at 22:20
  • $\begingroup$ Oh ok, I thought I saw something about putting the origin of the question. Sorry about that, I'm new here as well so not 100% sure what's accepted and what's not. $\endgroup$ Feb 4 at 22:20
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enter image description here

$CM // AD$. Extend it to meet $AB$ in $E$. Alternate interior angles $\angle MBE=\angle MDC$ which is equal to $\angle MBC$. Thus $\triangle MBC \cong \triangle MBE$. So $M$ bisects $CE$.

$AECD$ is a parallelogram. So $AD=CE$. We conclude $\triangle ADI \sim \triangle CMI$ with similarity ratio $2$.

Thus $MI=11\Rightarrow ID=22 \Rightarrow MD=33$.

From Pythagoras, you can find $CM$ and $AD=2CM$.

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  • $\begingroup$ So just checking, would $4\sqrt{190}$ would be the correct answer? $\endgroup$ Feb 4 at 22:57
  • $\begingroup$ @DarrenYang Yes I got same. $\endgroup$
    – cosmo5
    Feb 4 at 22:57
  • $\begingroup$ I've already upvoted, but it appears my upvote isn't counted to the publically displayed score due to <15 reputation. $\endgroup$ Feb 4 at 22:58
  • $\begingroup$ Ah I was suspecting that. No problem. $\endgroup$
    – cosmo5
    Feb 4 at 22:59

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