The Pythagorean Theorem In A Roof

The theorem can be proved algebraically using four copies of a right triangle with sides a a a b b b and c c c arranged inside a square with side c c c as in the top half of the diagram.

The pythagorean theorem in a roof.

What is the pythagorean theorem. You can also think of this theorem as the hypotenuse formula. When c pi 2 or 90 degrees if you insist cos 90 0 and the term containing the cosine vanishes. Since the roof forms an isosceles triangle the perpendicular from the top of the roof must divide the across length in two equal halves.

The roof pitch is the slope of the rafter. Let the length of one side of the roof be x ft. Pythagorean theorem the theorem states that. As usual ab c ac b bc a.

This theorem is talking about the area of the squares that are built on each side of the right triangle. It states that the area of the square whose side is the hypotenuse the side opposite the right angle is equal to the sum of the areas of the squares on the other two sides. It states that the sum of the squares of the sides of a right triangle equals the square of the hypotenuse. For example if a roof has a pitch of 4 12 then for every 12 inches the building extends horizontally it rises 4 inches.

The pythagorean theorem describes how the three sides of a right triangle are related in euclidean geometry. The pythagorean theorem is proven after two triangles are removed from each of the hexagons. In mathematics the pythagorean theorem also known as pythagoras s theorem is a fundamental relation in euclidean geometry among the three sides of a right triangle. A 2 b 2 c 2 proof of the pythagorean theorem using algebra.

Proof 39 by j. Define points d and e on ab so that ad ae b. Barry sutton the math gazette v 86 n 505 march 2002 p72 let in abc angle c 90 o. The pitch is commonly defined as the ratio of rise over run in the form of x 12.

The triangles are similar with area 1 2 a b frac 1 2 ab 2 1 a b while the small square has side b a b a b a and area b a 2 b. By construction c lies on the circle with center a and radius b. Pythagorean formula c 2 a 2 b 2 where a b are the legs of a right triangle and c is the hypotenuse. The square on the hypotenuse of a right triangle is equal to the sum of the squares on the two legs eves 80 81.

The pythagorean theorem is just a special case of another deeper theorem from trigonometry called the law of cosines c 2 a 2 b 2 2 a b cos c where c is the angle opposite to the long side c.