The Puzzles - Part 3

 What caused the sudden inclusion of geometry puzzles in the previous posts ?

The reason is that I noticed similar puzzles online on the web. However, the online site didn't provide the solution. Therefore, I looked carefully at the figures and noticed the concealed triangles with right angles included. That sent my mind to my high school math and out popped the famous Pythagoras ( pronounced Pie-tha-gora-s ) theorem!

I only remember a few things from my school and college days and this theorem happens to be one of the things I remember.

A Simple Puzzle: Part 2

 

If the inner square in yellow touches the outer rectangle in grey horizontally at 7 cms from the bottom right corner and vertically 2 cms from the bottom right as shown in the figure above, what is the area of the inner square?

Answer: 53 cm^2

Hint: see the method used in previous post.

A simple PUZZLE!

 Puzzle: If the area of the outer square is 100 then what is the area of the inner square?

The inner square touches the outer square at the mid-point of each of its four sides.

Answer: 

A square, by definition, has 4 equal sides and it's area is defined as the square of its side.

The outer square has an area of 100 which implies that each of its four sides is of length 10 because 10^2 is 100.

Pythagoras, from ancient Greece, came up with a relationship between the sides of a triangle that has a right angle between two of it's sides. The formula says:

 Hypotenuse ^2 = side1^2 + side2^2

(^2 is the sign for squaring)

Note: The hypotenuse is the longest side of the triangle opposite the right angle.

Why do we need this formula?

Because, we need to find the length of one side of the inner square. Each side of the inner square is a hypotenuse in the four triangles surrounding the inner square! Therefore the Pythagoras theorem applies. Let us choose the rightmost triangle at the bottom (see diagram above)

      Hypotenuse ^2 = 5^2 + 5^2

      Hypotenuse ^2 = 25 + 25 = 50.

5 is the length of each of the two sides that form a right angle in the triangle because the inner square touches the outer square at mid point of its sides.

And because hypotenuse is the same as a side of the inner square, the answer is that the area of the inner square is 50 !!