Pre-RMO 2014/3

Let ABCDABCD be a convex quadrilateral with perpendicular diagonals. If AB=20,BC=70,AB = 20, BC = 70, and CD=90CD = 90, then what is the value of DADA?

This note is part of the set Pre-RMO 2014

#Geometry #Pre-RMO

Note by Pranshu Gaba
6 years, 8 months ago

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1 vote

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Comments

Let the diagonals meet at a point EE, and let EA=a,EB=b,EC=c,ED=dEA=a, EB=b, EC=c, ED=d

Applying PT, we get

a2+b2=202→Eq.1a^{2}+b^{2}=20^{2} \rightarrow Eq.1

b2+c2=702→Eq.2b^{2}+c^{2}=70^{2} \rightarrow Eq.2

c2+d2=902→Eq.3c^{2}+d^{2}=90^{2} \rightarrow Eq.3

Eq.3 - Eq.2

d2−b2=902−702→Eq.4d^{2}-b^{2}=90^{2} -70^{2}\rightarrow Eq.4

Eq.1 + Eq.4

a2+d2=902−702+202a^{2}+d^{2}=90^{2} -70^{2}+20^{2}

⇒a2+d2=3600=AD2 \Rightarrow a^{2}+d^{2}=3600=AD^{2}

⇒AD=60 \Rightarrow AD=60

Aneesh Kundu - 6 years, 8 months ago

60

Rahul Verma - 6 years, 8 months ago

Its 60\boxed{60}

Akshat Sharda - 5 years, 10 months ago

60

Pooja Deshmukh - 6 years, 8 months ago

60

aditya vimal - 6 years, 8 months ago

60

Sahil Nare - 6 years, 1 month ago

60

Atharva Sarage - 6 years, 8 months ago

just a small correction a^2+b^2= 20^2

Aayush Patni - 6 years, 6 months ago

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Thnx edited.

Aneesh Kundu - 6 years, 6 months ago

60

gaurav singh - 5 years, 10 months ago

Ans underroot 140

Aman Real - 6 years, 8 months ago

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Only an integer answer is possible.

Sai Prasanth Rao - 6 years, 8 months ago
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