# 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
3 years, 11 months 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

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

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

$$\Rightarrow AD=60$$

- 3 years, 11 months ago

Its $$\boxed{60}$$

- 3 years, 1 month ago

60

- 3 years, 11 months ago

60

- 3 years, 5 months ago

60

- 3 years, 11 months ago

60

- 3 years, 11 months ago

60

- 3 years, 1 month ago

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

- 3 years, 10 months ago

Thnx edited.

- 3 years, 10 months ago

60

- 3 years, 11 months ago

Ans underroot 140

- 3 years, 11 months ago

Only an integer answer is possible.

- 3 years, 11 months ago