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.