A hidden Intersection!

Geometry Level 2

In a semicircle, a square is drawn with its diagonal perpendicular to the diameter. It is 1 unit away from the center and 4 units away from the circumference. What is the area of the orange square?


The answer is 12.

Mahdi Raza by Mahdi Raza , 16, India

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2 solutions

Saya Suka
Jun 7, 2021

Square's diagonal
= √((1 + 4)² – 1²)
= √24

Square's area
= ((Square's diagonal) / √2)²
= √24² / √2²
= 24 / 2
= 12

2 Helpful 0 Interesting 1 Brilliant 0 Confused
Pop Wong
Jun 7, 2021

Let x x be the side length of the square

The diagonal of the square will be x 2 + x 2 = 2 x \sqrt{x^2+x^2} = \sqrt{2} x

By Intersecting chords theorem , draw the lower half semi-circle with the same square,

2 x 2 = ( 5 + 1 ) × 4 ⟹ x 2 = 12 2x^2 = (5+1)\times4 \implies x^2 = \boxed{12}

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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