How can we explain the Pythagorean theorem

The Pythagorean theorem in words

With the Pythagorean Theorem you can make statements about the lengths of the sides and the squares over the sides of right triangles.

Terms in right triangles:

The hypotenuse is the longest side of the triangle, it is opposite the 90 ° angle.
The cathets are the shorter sides that are not opposite the 90 ° angle.

The Pythagorean theorem says:

Pythagorean theorem
In right triangles, the sum of the areas of the squares above the cathetus is the same as the area of ​​the square above the hypotenuse.

The Pythagorean theorem with variables

This is how the Pythagorean theorem can be described with variables:

If the $$ 90 ° $$ angle is $$ C $$, then the following applies: $$ c ^ 2 = a ^ 2 + b ^ 2 $$.

If the $$ 90 ° $$ angle is $$ B $$, then the following applies: $$ b ^ 2 = a ^ 2 + c ^ 2 $$.

If the $$ 90 ° $$ angle is $$ A $$, then the following applies: $$ a ^ 2 = c ^ 2 + b ^ 2 $$.

Count on Pythagoras

The set is suitable for calculating side lengths.

Example:

Using the Pythagorean Theorem, calculate the length of the side $$ c $$.

This is how you do it:

1. Can you use the sentence?

Yes, the triangle has a $$ 90 ° $$ angle.

2. Where is the right angle? What formula do you use

The $$ 90 ° $$ angle is $$ C $$. Then the formula is: $$ c ^ 2 = a ^ 2 + b ^ 2 $$

3. Put in the numbers.

$$ c ^ 2 = (6 $$ $$ cm) ^ 2 + (4 $$ $$ cm) ^ 2 $$

4. Rearrange the equation according to the quantity you are looking for and calculate.

$$ c ^ 2 = (6 $$ $$ cm) ^ 2 + (4 $$ $$ cm) ^ 2 = 52 $$ $$ cm ^ 2 $$ $$ | $$ take root

$$ c = sqrt (52 cm ^ 2) approx 7.2 $$ $$ cm $$


Or change first and then insert numbers:

$$ c ^ 2 = a ^ 2 + b ^ 2 $$ $$ | $$ take root

$$ c = sqrt (a ^ 2 + b ^ 2) = sqrt ((6 cm) ^ 2 + (4 cm) ^ 2) approx 7.2 $$ $$ cm $$

You can also change the formula first and then insert the numbers.

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Use in triangles with other names

Sometimes the triangles are not labeled with the letters $$ a $$, $$ b $$, $$ c $$. Then you look for the hypotenuse and then formulate the equation.

Example 1:

The $$ 90 ° $$ angle is opposite $$ t $$, so $$ t $$ is the hypotenuse. The equation is $$ t ^ 2 = s ^ 2 + u ^ 2 $$.

Example 2:

The $$ 90 ° $$ angle is opposite $$ t $$, so $$ t $$ is the hypotenuse. The equation is $$ t ^ 2 = s ^ 2 + u ^ 2 $$. Then you transform the equations according to the size you are looking for.

Calculation example

A $$ 255 $$ $$ m $$ - long rope was stretched from a tower at a distance of $$ 250 $$ $$ m $$. How high is the tower?

1. Can you use the sentence?

Yes, the tower is perpendicular to the earth.

2. Where is the right angle? What formula do you use

$$ h $$ and $$ e $$ enclose a right angle.
Then the formula is: $$ l ^ 2 = e ^ 2 + h ^ 2 $$

$$ e = 250 $$ $$ m $$, $$ l = 255 $$ $$ m $$, $$ l $$ is the hypotenuse.
You are looking for $$ h $$.

3. Put in the numbers.

$$ (255 $$ $$ m) ^ 2 = (250 $$ $$ m) ^ 2 + $$ $$ h ^ 2 $$

4. Rearrange the equation according to the quantity you are looking for and calculate.

$$ (255 $$ $$ m) ^ 2 = (250 $$ $$ m) ^ 2 + $$ $$ h ^ 2 $$ $$ | - (250 $$ $$ m) ^ 2 $$

$$ (255 m) ^ 2 - (250 m) ^ 2 = 2525 $$ $$ m ^ 2 = $$ $$ h ^ 2 $$ $$ | sqrt () $$

$$ h = sqrt (2525 m ^ 2) approx 50.25 $$ $$ m $$

The tower is $$ 50.25 $$ $$ m $$ high.


Or change first and then insert numbers:

$$ l ^ 2 = e ^ 2 + h ^ 2 $$ $$ | -e ^ 2 $$ then $$ | sqrt () $$

$$ h = sqrt (l ^ 2-e ^ 2) = sqrt ((255 m) ^ 2 - (250 m) ^ 2) $$

$$ h approx 50.25 $$ $$ m $$