Let the point E be the midpoint of the line

hyprkathknmk

hyprkathknmk

Answered question

2022-04-06

Let the point E be the midpoint of the line segment AD on the square ABCD. Then let a circle be determined by the points E, B and C as shown on the diagram. Which of the geometric figures has the greater perimeter, the square or the circle?

Answer & Explanation

recajossikpfmq

recajossikpfmq

Beginner2022-04-07Added 19 answers

Step 1
I will let the side of the square be 2. The center of the circle is at the intersection of the perpendicular bisectors of the chords BC and EB. Let that point be O and the foot of the perpendicular from O be F.
Now the triangle OEF is similar to EBA.
Therefore
O E E B = E F B A O E 5 = 5 / 2 2 O E = 5 / 4
The circumference of the circle is
2 π 5 4 = 5 π / 2 = 7.85
which is less than 8 the circumference of the square. They are very close.
I would like to see a proof that is not as computational as this one.
agrejas0hxpx

agrejas0hxpx

Beginner2022-04-08Added 4 answers

Step 1
Assume A B ¯ = 1 .
Let P be the intersection of the circle with AB.
Then A P ¯ × A B ¯ = A E ¯ 2 by power of point.
So A P ¯ = 1 / 4 , and P B ¯ = 3 / 4 . Hence the diameter of the circle is ( 3 / 4 ) 2 + 1 2 = 5 / 4 . The rest follows like the other proofs.

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