PYTHAGOREAN THEOREM (Proof by Rearrangement: Part 1)

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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.
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Let us make three more of these so we have four congruent right triangles.
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Now, let us arrange the four right triangles to form a square like this
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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. 
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Let us take note of that the ares of the inner square is c^2.
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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. 
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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.
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Let us focus on the area being left by the two triangles and shade it with white.
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If you notice, the white area can be divided into two like this
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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
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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. 
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This means that the upper side of the small quadrilateral measures a. Hence, the small quadrilateral is a square.
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The area of the small square is
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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.
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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.
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This means that the measure of the lower side of the big white quadrilateral is b. Hence, the big white quadrilateral is a square.
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The area of the big white square is
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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.
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Therefore, for any right triangles 


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PYTHAGOREAN THEOREM EXPLORATION 2 (CUT-OUTS)

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Pythagorean Theorem Exploration 2 Cutouts
Here is the second part of the previous post (Pythagorean Exploration 1 cut-outs). The first one uses the other two sides (a and b) of the right triangle to form the sides of the square pattern. This time, the hypotenuse (c) will be used for the sides of the square. You may use the discussion in Pythagorean Exploration 2 as a guide.

You may download this template for personal use and for your math class activity. The length of the sides on the second page measures 6.5 inches. This is also the measure of the length of the hypotenuse of the right triangle on the first page.



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PYTHAGOREAN THEOREM EXPLORATION 1 (CUT-OUTS)

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Pythagoras Theorem Cutouts
In the previous posts, two different explorations were presented for the derivation of the Pythagoras Theorem. The first exploration can be found here and the second here.

The Pythagoras Theorem states that:
   "The square of the length of the hypotenuse (denoted by c) is equal to the sum of the squares of the lengths of the other two sides (denoted by a and b)"
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Pythagoras Theorem
This post is for the downloadable cutouts for each of the explorations. You may download them and use them for your class or for your kids to learn.

Here is the cutout for the first exploration of the Pythagoras Theorem. The file consist of two pages. The first page consists of four congruent right triangles. The second page is a square.

Cut along the broken lines on the right triangles and use them to fill in the square on the second page using the EXPLORATION 1 as a guide.

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PYTHAGOREAN THEOREM (Exploration 2)

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The Pythagorean Theorem
Here is another proof for the Pythagorean Theorem. You can see the first part here.
Start with a cutout of a right triangle.
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Cut three more cutouts.
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Let us name the sides of the right triangle with c as the hypotenuse.
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Now, let us form a square using the four (4) triangles. This time, use the hypotenuse as the sides of the square.
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Let us take a look at the different parts of the square we formed. If we separate the whole square, we can get its area in terms of c.
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The middle part is also a square with sides equal to the difference of sides b and a. The area can be obtained as
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On the other hand, the area of the four triangles outlining the sides of the whole square is as follows.
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Combining their areas, we obtain
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Thus, c^2 = a^2 + b^2.

Your questions, comments and suggestions are welcome here. Kindly write them in the comment box below.

PYTHAGOREAN THEOREM (Exploration 1)

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The Pythagoras' Theorem
The Pythagorean theorem is also known as the Pythagoras' theorem. It is named after a famous Greek mathematician, Pythagoras, who is credited for its proof.

The theorem states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the remaining sides. That is c^2 = a^2 + b^2, where c is the hypotenuse, a and b are the lengths of the remaining sides.

Here is one of the theorem. Take note that the theorem only applies to right triangles.
Start with a cutout of any right triangle.
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 Let us name the shortest side as a, the longer side as b and the hypotenuse as c.
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Let us make three (3) more of the right triangle.
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Our next step is to form a square out of the four (4) triangles, making sure that the combined length of a and b will be the length of the side of the square that we are forming.
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Let us take a look at the different parts of the square we formed. If we separate the whole square, we can get its area in terms of a and b.
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The middle part is also a square with sides equal to the hypotenuse of the right triangle. The area can also be obtained.
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The area of the four right triangles outlining the sides of the whole square is as follows
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We take note that the area of the whole square is equal to the area of the small square in the middle added to the area of the four right triangles. That is
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Thus, c^2 = a^2 + b^2.

Hope this will help you understand the derivation of the Pythagoras' Theorem.

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EXTERIOR ANGLE THEOREM (Part 2: Proof)


Here is the two-column proof for the EXTERIOR ANGLE THEOREM. You may download and print it for your perusal.


Your comments and suggestions are welcome here. You may write them in the comment box below. 

EXTERIOR ANGLE THEOREM (Part 1: Exploration)

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Many are already familiar with the angles of a triangle. Many can easily point out which are the interior angles and even draw the exterior angles.

On the other hand, how many are you are familiar with the exterior angle theorem? What is the relationship of the exterior angles with the interior angles? The answer for these questions can be summarized by the "Exterior Angle Theorem".

In this post, let us discover where this theorem come from by using some illustrative examples.

Materials needed:  ruler, protractor, calculator and pencil

Procedure:
1) On a piece of paper, draw one (1) acute triangle, one (1) right triangle, and one (1) obtuse triangle. Name each triangle as triangle ABC, with AC as the base.
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2) Extend the base of AC to a point D. Points A, C and D should be on the same line and C is between A and D.
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  • angle BCD is the exterior angle of each triangle
  • angles A and B are the remote (non-adjacent) interior angles of each triangle
3) Measure each of the angles and complete the table below:
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4) Look at the rightmost part of the table. Compare the sum of the measures of angles A and B and the exterior angle. 
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5) We can now make a conclusion/generalization about the relationship of the measurement of the exterior angle and the remote interior angles.
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Here is the pdf copy of the activity, if to be conducted in a classroom setting. You are free to download it and use for your class. Hope it will be beneficial for you.

Your comments and suggestions are accepted here. Just write them in the comment box below. Thank you!  



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