# Pre-RMO 2014/3

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

This note is part of the set Pre-RMO 2014 Note by Pranshu Gaba
6 years, 1 month ago

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

Applying PT, we get

$a^{2}+b^{2}=20^{2} \rightarrow Eq.1$

$b^{2}+c^{2}=70^{2} \rightarrow Eq.2$

$c^{2}+d^{2}=90^{2} \rightarrow Eq.3$

Eq.3 - Eq.2

$d^{2}-b^{2}=90^{2} -70^{2}\rightarrow Eq.4$

Eq.1 + Eq.4

$a^{2}+d^{2}=90^{2} -70^{2}+20^{2}$

$\Rightarrow a^{2}+d^{2}=3600=AD^{2}$

$\Rightarrow AD=60$

- 6 years, 1 month ago

60

- 6 years, 1 month ago

Its $\boxed{60}$

- 5 years, 3 months ago

60

- 6 years, 1 month ago

60

- 6 years, 1 month ago

60

- 5 years, 7 months ago

60

- 6 years, 1 month ago

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

- 6 years ago

Thnx edited.

- 6 years ago

60

- 5 years, 3 months ago

Ans underroot 140

- 6 years, 1 month ago

Only an integer answer is possible.

- 6 years, 1 month ago