A geometry problem by Ronak Singha

Geometry Level 3

Point A is inside the square BCDE whose side length is 20. The length of AB is 9 and the length of AE is 13. Find x the length of AC.

Take the approximate value


The answer is 17.4.

Ronak Singha by Ronak Singha , 27, India

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1 solution

Nguyen Thanh Long
Jul 23, 2015

Applying Heron for triangles ABE theory we have: denote: P = A B + B E + A E 2 P=\frac{AB+BE+AE}{2} [ A B E ] = P × ( P − A B ) × ( P − B E ) × ( P − A E ) [ABE] = \sqrt{P \times (P-AB) \times (P-BE) \times (P-AE)} Denote AH is foot of the line goes through B and orthogonal with BE. → A H = 2 × [ A B E ] B E \rightarrow AH=\frac{2 \times [ABE]}{BE} A C = ( 20 − A H ) 2 + ( 9 2 − A H 2 ) = 17.4 AC = \sqrt{(20-AH)^2+(9^2-AH^2)}=\boxed{17.4}

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