Pythagorean theorem is the theorem of geometry establishing connection between the parties of a rectangular triangle. The theorem is a statement for which in the considered theory there is a proof. At the moment there are more than 300 ways of the proof of Pythagorean theorem, however as the Basic Element of the school program is used the proof through similar triangles.

## It is required to you

- the page of a notebook in a cage
- ruler
- pencil

## Instruction

1. Pythagorean theorem sounds as follows: rectangulartrianglethe square of a hypotenuse is equal in to the sum of squares of legs. The geometrical formulation demands also a concept of the area: the area of the square constructed on a hypotenuse is equal in a rectangular triangle to the sum of the areas of the squares constructed on legs.

2. Draw a rectangular triangle with tops of A, B, C where a corner of C – a straight line. The party of BC designate a, the party of AC designate b, designate the party of AB with.

3. Carry out height from a corner of C and designate its basis through H. Triangles are similar if two corners of one triangle are respectively equal to two corners of other triangle. H corner – a straight line, as well as a corner of C. Therefore, the triangle of ACH is similar to a triangle of ABC on two corners. The triangle of CBH is also similar to a triangle of ABC on two corners.

4. Work out the equation where a belongs to c as HB belongs to and. Respectively, b belongs to c as AH belongs to b.

5. Solve these equations. To solve the equation, multiply numerator of the right fraction by a denominator of the left fraction, and a denominator of the right fraction – by numerator of the left fraction. We receive: an in a square = sHB, b in a square = cAH.

6. Put these two equations. We receive: an in a square + b in a square = c (HB + AH). As HB + AH = c, as a result has to turn out: an in a square + b in a square = with in a square. This completes the proof.