Regular pentagon ABNER?

Question

Regular pentagon ABNER and parallelogram NERD. Prove that A, N, D are collinear, and prove that N is the midpoint of AD.

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Answer 1

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Parallelogram Nerd is a rhombus specifically!!!!
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All angles of regular pentagon are 108°
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Join A and D
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Therefore ________________________________________
Also ________________________________________
Therefore ________________________________________
=108° - 72°
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= 36°
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Since EN = RD (Opp sides of a parm are equal)
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and EN = RA (sides of a regular pentagon are equal)
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Δ ARD is isosceles
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Therefore ________________________________________
Δ are wqual)
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Since the angle sum of a Δ = 180°
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= 180°
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Thus ________________________________________
ie A, D and N are collinear __________________QED
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N cannot be the midpoint of AD as N is external to AD .... D
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cannot even be the midpoint of AN.
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Proof
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AN < BN + AB (Triangle inequality)
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= RE + RE (Since all sudes of regular pentagon are equal)
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= 2RE
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Now RE = DN
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So AN <2DN
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ie DN > ½AN
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So D is NOT the midpoint of AN
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Hope that this helps ^_^!!!

Answer 2

(Actually NERD is a rhombus!)

All angles of regular pentagon are 108°

Join A and D
Therefore Also Therefore =108° - 72°
= 36°
Since EN = RD (Opp sides of a parm are equal)
and EN = RA (sides of a regular pentagon are equal)
Δ ARD is isosceles
Therefore Since the angle sum of a Δ = 180°
= 180°
Thus ie A, D and N are collinear __________________QED

N cannot be the midpoint of AD as N is external to AD .... D cannot even be the midpoint of AN.

Proof

AN < BN + AB (Triangle inequality)
= RE + RE (Since all sudes of regular pentagon are equal)
= 2RE

Now RE = DN
So AN <2DN
ie DN > ½AN
So D is NOT the midpoint of AN