PYTHAGOREAN THEOREM (Proof by Rearrangement: Part 1)

Sunday, January 25, 2015

IGCSE,Pythagoras,right triangles,math proof,math explorations,mathematics,geometry

Here is another proof of the Pythagorean theorem.
Let us use a right triangle and name the shortest side as a, the longer side as b, and the hypotenuse as c.

Let us make three more of these so we have four congruent right triangles.

Now, let us arrange the four right triangles to form a square like this

In this figure, there are two squares formed. The first square is larger square, with side equal to a+b, while the second square is the inner square with side equal to c.

Let us focus on the inner square. The length of its side is equal to the hypotenuse of the four right triangles. It means that it has sides each measuring as c. Hence, the area of the square is c^2.

Let us take note of that the ares of the inner square is c^2.

Now, let us label the four right triangles as triangle 1, triangle 2, triangle 3 and triangle 4. This will make it easier for us to identify which triangle is moved later.

Let us rearrange the triangles. Let us move triangle 2 beside triangle 1, and triangle 4 beside triangle 3. In this case, each pair will form a rectangle.

Let us focus on the area being left by the two triangles and shade it with white.

If you notice, the white area can be divided into two like this

We formed two quadrilaterals but we are not yet sure if they are squares or not.

The smaller quadrilateral has a side that is equal to the shortest side of triangle 4. This means that this side measures a.

On the other hand, if we slide back triangle 2, we could see that the upper side of the small quadrilateral is also the shortest side of triangle 2.

This means that the upper side of the small quadrilateral measures a. Hence, the small quadrilateral is a square.

The area of the small square is

Now, let us look at the bigger quadrilateral shaded with white. One of its side (leftmost) has the same length as the longer side of triangle 2. It means that this side measures b.

On the other hand, if we slide back triangle 4, we could see that its longer side coincides with the lower side of the big white quadrilateral.

This means that the measure of the lower side of the big white quadrilateral is b. Hence, the big white quadrilateral is a square.

The area of the big white square is

If we compare the two figures formed. Both of them has four (4) right triangles and the areas of these triangles are the same. It means that the area of the white inner square in the first figure is the same as the area of the two white squares in the second figure.

Therefore, for any right triangles

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{ 5 comments... read them below or Comment }

WDxdFebruary 25, 2016 at 2:16 AM
This helped a lot. Thank you and may God bless you too.
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OscarNovember 29, 2019 at 2:20 AM
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OscarNovember 29, 2019 at 2:21 AM
Thanks for the tips and illustration. I'll use it at my tutorial center.
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Stabilit AmericaAugust 18, 2023 at 4:17 AM
The Pythagorean theorem is a fundamental mathematical tool that is applied in many fields, including architecture and civil engineering. Although it is not directly used to construct a building, it can be used in calculations related to the geometry and structure of the building. Here are some ways in which the Pythagorean theorem could be applied in the construction of a building:

Verification of right angles: In construction, it is essential to ensure that corners and structures have precise right angles. The Pythagorean theorem can be used to check if a triangle is a right triangle, which in turn can help ensure perfectly perpendicular corners and structures.

Distance measurements: In the design and planning phase, the Pythagorean theorem can be used to calculate diagonal distances. This can be useful for determining the length of diagonals in architectural plans, such as the length of a diagonal wall or the distance between two points on a piece of land.

Structural calculations: In structural engineering, the Pythagorean theorem can be applied to calculate the lengths of members in a structural system. For example, when designing support systems, such as beams and columns, calculations based on the Pythagorean theorem can be used to ensure that the elements are strong enough to support the required loads.

Leveling and alignment: In construction, it is essential to correctly level and align the elements to ensure the stability and safety of the building. The Pythagorean Theorem can be used to make leveling measurements and to ensure that structures are at precise angles.

Stair and Ramp Design: When designing stairs or ramps in a building, the Pythagorean theorem can be used to calculate the dimensions needed to achieve certain slopes or step heights.

Importantly, while the Pythagorean theorem is a valuable tool in construction, it is used as an underlying mathematical tool to solve distance and geometric problems in the context of architecture and civil engineering. Practical applications may vary depending on the phase of the project and the specific needs of the building design and construction.
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