Show how each of these pairs of polygons can be decomposed i

Show how each of these pairs of polygons can be decomposed into congruent polygons:

(a) An isosceles right triangle and a square.

(b) A 1x1 square and a rectangle with one side equal to root(2) .

(c) An equilateral triangle with each side of length 1 and an isosceles triangle with base of length 1/2.

Solution

A right isoscles triangle can be decomposed into a square by making a cut from the vertex on the base of triangle to the mid point of the hyposteuse and then attaching the cut part to remaining part along the base.

2. A square and a rectangle.

Make a horizontal cut on the sqaure at a distance of 0.41 unit from top .Remove this part.Make a cut of 0.69 from the side of 1 unit on the cut part. Attach it to the side of the original sqaure which is now a rectangle of height 0.59 unit. attach it by the side. Thus the new shape will be a rectangle of 1.41 unit length and 0.59 unit breath.

Note that we are still left with addition sqaare sheet of 0.41 units sde.Cut it into strips of 0.1 unit breath and 0,41 length and attach it on the top of the rectangle. Ths will add a layer of breath 0.1 and length 1.41 on the rectangle.

Repeat it till the additonal paper is not finished.

The rectangle from sqaure will be obtained thus.

3. Decomposition of an equilateral triangle inot isosceles triangle

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