## Sunday, March 5, 2023

### Geometry Problem 1520: Discovering Distances in a Rectangle with an Exterior Point: A Geometry Challenge. Difficulty Level: High School.

Geometry Problem 1520. Post your solution in the comment box below.
Level: Mathematics Education, K-12 School, Honors Geometry, College.

Details: Click on the figure below.

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1. 12V2 (V2 = sqrt of 2)

2. Complete Right Triangle BEF, F on BC extended.
Complete Right Triangle DEG, G on DC extended.
Let AD = a and CD = b

Triangles DEG & ACD are congruent ASA
So GE = b = CF and hence EF = a - b and BF = a + b

So in Triangle BEF,
BE^2 = BF^2 + EF^2 = (a+b)^2 + (a-b)^2 = 2 (a^2 + b^2) = 2 AC^2 = 2 X 12^2

Therefore BE = 12. sqrt2

Sumith Peiris
Moratuwa
Sri Lanka

3. Connect B and D then <BDE=90⁰
BE² = BD² + DE²
BE² = 12² + 12² = 12 ROOT2