Find The Rectangle

Geometry Level pending

Let A B C D ABCD be a quadrilateral where A B = x AB = x , B C = x + 1 BC = x+1 , A C = x + 2 AC = x+2 , and point D D is found by reflecting B B about the diagonal A C ‾ \overline{AC} . Find the value of x x such that A B C D ABCD is a rectangle.


The answer is 3.

Daniel Hinds by Daniel Hinds , 20, United Kingdom

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1 solution

Since the quadrilateral A B C D ABCD is a rectangle, we have ( ∣ A C ‾ ∣ ) 2 = ( ∣ A B ‾ ∣ ) 2 + ( ∣ B C ‾ ∣ ) 2 (|\overline {AC}|)^2=(|\overline {AB}|)^2+(|\overline {BC}|)^2 or x 2 + ( x + 1 ) 2 = ( x + 2 ) 2 ⟹ x 2 − 2 x − 3 = 0 x^2+(x+1)^2=(x+2)^2\implies x^2-2x-3=0 . Since x x is positive, it is the positive root of this equation, which is 3 \boxed 3 .

1 Helpful 0 Interesting 0 Brilliant 0 Confused

D is not found by reflecting B in the diagonal AC!

Dick van der Leeden - 1 year, 2 months ago

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