The Pythagorean Theorem Diagrams is used to find the length of one of the sides of a right triangle knowing its other two sides. This theorem has been of great aid to mankind in many areas. In the area of architecture, this theorem is used in building design and construction, for example to calculate the length of a right-triangular truss.
Pythagorean theorem and its proof are closely related to the areas of a square and triangle. Therefore, to understand Pythagorean theorem you have to have knowledge of finding the areas of a square and triangle. To refresh your memory on the formulas for the area of a square and a right triangle, see the following description!
1. The area of ABCD square = AB x BC = AB x AB (since AB = BC) = AB². So the area of square = side length x side length.
2. The area of a Right Triangle ABC = 1/2 x AB x AC.
If AB and AC are the legs of the triangle, then the area of the right triangle = 1/2 x length of leg I x length of leg II
A right triangle has two legs or catheti (singular:cathetus) and hypotenuse. The first figure above is a right triangle ABC having a right angle at the vertex A. The two sides that meet at a right angle, namely AB and AC are called its legs. The side opposite to the right angle, namely BC is called its hypotenuse.
BC² = AC² + AB² ,or
a² = b² + c²
Look at the diagram above!
Figure (i) and (ii) above show two squares whose sides have equal length of (b+c). Having equal side length, the areas of both squares must also be equal. The shaded region in figure i and ii are equal in area. This in turn shows that the non-shaded regions are also equal in area. Hence, a² = b² + c².
Next look at the figure (iii). The figure is constructed from rearranging pieces in figure i and ii. The area of the square on the hypotenuse is a² and the sum of the areas of the squares on the legs is (b² + c²).
From the discussion above, for every right triangle = the area of the square on the hypotenuse equals the sum of the areas of the squares on the other two sides (legs)
Let’s practice and find answer the question in picture above!
p² = 15² – 12²
p² = 255 – 144
p² = 81
p = √81 = 9
Thus, p = 9