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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. What is tangent of angle forming by one side of bigger circle and line between hanging corner and one corner of bigger square as shown in figure.
Assume lengths of sides of triangle formed at top right of the figure are 'a' and 'b' as shown in figure.
Step 2:
Construct some similar triangles to top right triangle as shown in figure.
For better understanding I given names to all triangles.
Step 3:
Triangles 2, 3, 4, 5 are similar so their lengths are also similar.
In triangle 1: two angles are 45 so it's two sides are same.
Step 4:
Observe triangle 5:
tan α = Opposite side / Adjacent side
Opposite side = (2b+a)-a-b = b
Adjacent side = (2b+a)-a = 2b
⇒ tan α = b/2b
⇒ tan α = 1/2
