Pythagoras theorem

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The Pythagorean theorem is a rule that is fulfilled in the case of a right triangle, the sum of each of the legs squared being equal to the hypotenuse squared.

We must take into account that this law is only fulfilled for a very particular type of triangle, the right triangle, which is one where two of the three sides, which are called legs, form a right angle, that is, they measure 90º.

We can observe the Pythagorean theorem in the following formula, where AB and BC are the legs and AC is the hypotenuse of the triangle shown in the graph below.

AB2 + BC2 = AC2

So, the Pythagorean theorem allows us to calculate the length of one of the sides of the triangle when we know the other two. Likewise, knowing the length of all the sides, we can verify without a triangle it is right.

It should be noted that in the figure shown the angle measurements are referential. They can have different measures, but in all triangles, in general (not only in rectangles), the interior angles must always add up to 180º. Therefore, if one measures 90º, the sum of the other two must necessarily be 90º.

So, taking into account the above, in a right triangle one of the angles is right and the other two must be acute (less than 90º).

Example of application of the Pythagorean theorem

Suppose we have a right triangle, the length of its hypotenuse being 15 meters and that of one of its legs 10 meters. How long is the other leg?

So, we develop the operation:

152 = 102 + x2

225 = 100 + x2

x2 = 125

x = 11.1803 meters

Let's look at another exercise. You could tell us that you have a triangle whose sides are 8, 11 and 14 meters. Can it be a right triangle?

82+112=64+121=185

142=196

185 ≠ 196

Therefore, the triangle cannot be right (at this point it should be noted that the hypotenuse will always measure more than the legs).

Now, as a third example of applying this theorem, suppose we are told that we have a square whose sides are 12 meters. What is the length of its diagonal?

In this case, we must remember that the interior angles of a square measure 90º. Therefore, when we draw a diagonal we divide the figure into two right triangles (as seen in the figure below).

So the length of the diagonal (x) would be:

122 + 122 = x2

144 + 144 = x2

x2 = 288

x = 16.9706 meters

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