Pythagoras theorem

Pythagoras theorem

Hello Science Bee, we will discuss the pythagoras theorem here.

 

What is the Pythagoras theorem?

The Pythagorean theorem, also known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The theorem can be written as an equation relating the lengths of the sides a, b and c, often called the "Pythagorean equation": 
where c represents the length of the hypotenuse and a and b the lengths of the triangle's other two sides.
As pointed out in the introduction, if c denotes the length of the hypotenuse and a and b denote the lengths of the other two sides, the Pythagorean theorem can be expressed as the Pythagorean equation: 
If the length of both a and b are known, then c can be calculated as 
If the length of the hypotenuse c and of one side (a or b) are known, then the length of the other side can be calculated as 
or 

Video: Visualizing Pythagoras Theorem
 
The Pythagorean equation relates the sides of a right triangle in a simple way, so that if the lengths of any two sides are known the length of the third side can be found. Another corollary of the theorem is that in any right triangle, the hypotenuse is greater than any one of the other sides, but less than their sum.
A generalization of this theorem is the law of cosines, which allows the computation of the length of any side of any triangle, given the lengths of the other two sides and the angle between them. If the angle between the other sides is a right angle, the law of cosines reduces to the Pythagorean equation.

 

Solving Pythagoras theorem word problem

This video shows how to solve the Pythagoras theorem word problems.

 

Proof using similar triangles

Proof using similar triangles
 
This proof is based on the proportionality of the sides of two similar triangles, that is, upon the fact that the ratio of any two corresponding sides of similar triangles is the same regardless of the size of the triangles.

Video: How to prove Pythagoras's Theorem in under 1 minute.
 
Let ABC represent a right triangle, with the right angle located at C, as shown on the figure. Draw the altitude from point C, and call H its intersection with the side AB. Point H divides the length of the hypotenuse c into parts d and e. The new triangle ACH is similar to triangle ABC, because they both have a right angle (by definition of the altitude), and they share the angle at A, meaning that the third angle will be the same in both triangles as well, marked as θ in the figure. By a similar reasoning, the triangle CBH is also similar to ABC. The proof of similarity of the triangles requires the triangle postulate: the sum of the angles in a triangle is two right angles, and is equivalent to the parallel postulate. Similarity of the triangles leads to the equality of ratios of corresponding sides:
The first result equates the cosines of the angles θ, whereas the second result equates their sines.
These ratios can be written as 
Summing these two equalities results in 
which, after simplification, expresses the Pythagorean theorem: 

Video: Origami Proof of the Pythagorean Theorem

 

Solving for the converse of the Pythagoras theorem

This video shows how to solve the Pythagoras theorem word problems.

 

Solving Pythagoras theorem and radicals

This video shows how to solve the Pythagoras theorem and radicals.

 

How to find the Pythagorean triples/triads 

This video shows how To find The Pythagorean Triples/Triads 

 

How to visualize all possible Pythagorean triples?

This video shows how To visualize the Pythagorean Triples/Triads. 

 

Law of cosines

The separation s of two points (r1, θ1) and (r2, θ2) in polar coordinates is given by the law of cosines. Interior angle Δθ = θ1−θ2.
The Pythagorean theorem is a special case of the more general theorem relating the lengths of sides in any triangle, the law of cosines:
where θ is the angle between sides a and b.
When θ is 90 degrees (π/2 radians), then cosθ = 0, and the formula reduces to the usual Pythagorean theorem.

Video: Law of Cosines

 


Let's Review

  1. The ____ theorem, also known as ____ ' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The theorem can be written as an equation relating the lengths of the sides ab and c, often called the "Pythagorean equation": 

     where c represents the length of the ____  and a and b the lengths of the triangle's other two sides.

  2. If the length of both a and b are known, then c can be calculated as ____ ,

  3. If the length of the hypotenuse c and of one side (a or b) are known, then the length of the other side can be calculated as 

    ____  or  ____ 

  4. This proof is based on the proportionality of the sides of two similar triangles, that is, upon the fact that the ____ of any two corresponding sides of similar triangles is the same regardless of the size of the triangles.

  5.  

     

    The Pythagorean theorem is a special case of the more general theorem relating the lengths of sides in any triangle, the law of cosines:

     where θ is the angle between sides a and b.

    When θ is 90 degrees (π/2 radians), then cosθ = 0, and the formula reduces to the usual ____ theorem.

     

    Answer

    1.  The Pythagorean theorem, also known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The theorem can be written as an equation relating the lengths of the sides ab and c, often called the "Pythagorean equation": 

    2.  where c represents the length of the hypotenuse and a and b the lengths of the triangle's other two sides.

    3. If the length of both a and b are known, then c can be calculated as 

    4. If the length of the hypotenuse c and of one side (a or b) are known, then the length of the other side can be calculated as 

       or  

    5. This proof is based on the proportionality of the sides of two similar triangles, that is, upon the fact that the ratio of any two corresponding sides of similar triangles is the same regardless of the size of the triangles.

    6. The Pythagorean theorem is a special case of the more general theorem relating the lengths of sides in any triangle, the law of cosines:

       where θ is the angle between sides a and b.

      When θ is 90 degrees (π/2 radians), then cosθ = 0, and the formula reduces to the usual Pythagorean theorem.

     

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