Geometry Puzzle: If the Area of the Blue Square is 25 Find the Area of the Red Square

🔷 Welcome to Geometry Puzzle series! In this brain-teasing video, we present a captivating puzzle: If the area of the blue square is 25 square units, can you find the area of the small red square? 🟥
🧩 Challenge your geometric reasoning and problem-solving skills as we delve into this intriguing puzzle. We'll do step by step solution together. 💡
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[@larswilms8275]

General solution:
area of large square = a^2
area of smaller square = x^2
then
(a/2 + x)^2 + (x/2)^2 = (a√2/2)^2 (expand the squares)
(a^2) / 4 + a x + x^2 + (x^2) / 4 = 2 * (a^2) / 4 (multiply by 4)
a^2 + 4 a x + 5 x^2 = 2 a^2 (subtract 2 a^2)
5 x^2 + 4 a x - a ^2 = 0 (abc quadratic equation)
x(1,2) = (-4a +/- √((4a)^2 - 4 * 5 * (-a^2)))/ (2 * 5) = (-4a +/- √(16a^2 + 20 a^2))/ 10 = (-4a +/- √(36a^2))/10
x = (-4a + 6a)/10 or x = (-4a - 6a)/10
x = 2a/10 or x = -10a/10
x= a/5 or x = -a
The second answer (x = -a) would give us the blue starting square, so that kind of makes sense.
The first answer shows that there will always be a 5 times reduction in side of the square, and consequently a 25 times reduction in area of the smaller square.

[@GourawRaj-hj4jy]

Cna we say that the area of left and right white part in lower of side of blue one is equal??
If yes then answer might be different

[@nagarajabillurrao]

Small squares are equal. Side equal to half of radius dia - 5. Any flaw in my argument?

[@prossvay8744]

Red area =1^2=1 square units

[@cosmolbfu67]

Chord theorem
Let side of the small square = x
x(5+x)=(2.5-x/2)(2.5+x/2)
x(5+x)=1/2(5-x)1/2(5+x)
4x(5+x)=(5-x)(5+x)
20x+4x^2=25-x^2
5x^2 + 20x - 25 =0
x=1
A=x^2=1 ***

[@JobBouwman]

Call s the side of the small square, then Thales and Pythagoras tell us that:
(5 + 2s)^2 + s^2 = (5*sqrt(2))^2
5s^2 + 20s -25 =0
5(s+5)(s-1) = 0
Hence s = 1

[@nenetstree914]

Area is 1.

[@sergeyvinns931]

А=1.