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Proof using similar triangles





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Proof using similar triangles

This proof is based on the proportionality of the sides of two similar triangles, that is, upon the fact that the ratio of any two corresponding sides of similar triangles is the same regardless of the size of the triangles.

Let ABC represent a right triangle, with the right angle located at C, as shown on the figure. Draw the altitude from point C, and call H its intersection with the side AB. Point H divides the length of the hypotenuse c into parts d and e. The new triangle ACH is similar to triangle ABC, because they both have a right angle (by definition of the altitude), and they share the angle at A, meaning that the third angle will be the same in both triangles as well, marked as θ in the figure. By a similar reasoning, the triangle CBH is also similar to ABC. The proof of similarity of the triangles requires the Triangle postulate: the sum of the angles in a triangle is two right angles, and is equivalent to the parallel postulate. Similarity of the triangles leads to the equality of ratios of corresponding sides:

8. Now we would like to offer to your attention to the proof by rearrangement. We have already said some words about the Pythagorean proof which was a proof by rearrangement and now the same idea will be presented by the animationwhich consists of a large square, side a + b, containing four identical right triangles. The triangles are shown in two arrangements, the first of which leaves two squares a2 and b2 uncovered, the second of which leaves square c2 uncovered. The area encompassed by the outer square never changes, and the area of the four triangles is the same at the beginning and the end, so the black square areas must be equal, therefore a2 + b2 = c2.

9.In the second part of our presentation we would like to draw your attention how and where the Pythagorean Theorem is used today. It is used often in construction, in engineering, in architecture, in design, in art and in aeronautics.

The most famous application of the theorem is to find the shortest distance between two points.Physicists and engineers use the Pythagorean Theorem all the time because right triangles are everywhere. For example, on a map, calculating the high of an object.

Architects use the Pythagorean Theorem, which is expressed by the equation: a2 + b2 = c2, in designing and computing the measurements of building structures and bridges.

10. We would like to show a video for those people who like playing football could be interesting to get to know that a football peach is designed on the basis of the Pythagorean Theorem.

11.We live an amazing world who could ever think thet in the 21 st century people will apply the Pythagorean Theorem whivh goes back 600-500 B.C.

To compose this presentation we have applied the following sources:

Thanky for your attention!

 

 

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