Hard looking Easy Puzzle | One square is inscribed in other square and point of intersection divides line in 3:2 ratio | Find area of square

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Take a look at figure given below. Give it a try and when you are ready then watch the solution.

Two squares are drawn in such a way that one square is inscribed in other square. One corner is on diagonal of bigger square, two corners on two adjacent sides and one is hanging inside diagram. One corner on the side divides side of bigger square in 3 and 2 length. Find area of square.

As you can see, side length must be 5 unit.

Let side of square is 's'.

⇒ We have to find value of s²

Solution:

Step 1:

Give names to all points.

Step 2:

Draw one perpendicular line on AD passing through F as shown in figure below.

Step 3:

In triangle FPE, and EDH.

You can see both have one side same and one angle is 90.

And also angle formed on either side of point E is are total of 90.

So these triangle must be similar.

⇒ PF = ED = 2

⇒ PE = DH = √(s²-4)

Step 4:

In triangle APF, two angles are 45

⇒ AP = PF

AP = 3 - √(s²-4)

PF = 2

⇒ 3 - √(s²-4) = 2

⇒ √(s²-4) = 1

⇒ s²-4 = 1

⇒ s² = 5

So Area of Small Square is 5 square units and Side is √5