Pythagoras’ Theorem and Proof

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What is Pythagoras (Pythagorean) Theorem for triangles? The proof of Pythagorean theorem and examples.

Pythagorean Theorem;

Pythagorean-theorem-1 The diagram shows ΔABC which is right angled at A and [AD] is an altitude, that is [AD] ⊥ [BC]

Let \displaystyle m\overset{\wedge }{\mathop{B}}\,=\alpha

\displaystyle \Rightarrow m\overset{\wedge }{\mathop{BAD}}\,=\left( 90-\alpha \right)

\displaystyle \Rightarrow m\overset{\wedge }{\mathop{DAC}}\,=90{}^\circ -\left( 90{}^\circ -\alpha \right){}^\circ =\alpha {}^\circ \left( m\overset{\wedge }{\mathop{BAC}}\,=90{}^\circ \right)

\displaystyle \Rightarrow m\overset{\wedge }{\mathop{C}}\,=\left( 90-\alpha \right){}^\circ

So we now have three equiangular triangles ABC, DBA and DAC each having angles measuring :

90°, α° and (90 – α)°

So ΔABC ∼ ΔDBA ∼ ΔDAC

If we take ΔABC ∼ ΔDBA, one of the proportions is

\displaystyle \frac{\left| AB \right|}{\left| DB \right|}=\frac{\left| BC \right|}{\left| BA \right|}

OR \displaystyle \frac{c}{p}=\frac{a}{c}\Rightarrow {{c}^{2}}=ap

If we take ΔABC ∼ΔDAC, one of the proportions is

\displaystyle \frac{\left| AC \right|}{\left| DC \right|}=\frac{\left| BC \right|}{\left| AC \right|}

OR \displaystyle \frac{b}{k}=\frac{a}{b}\Rightarrow {{b}^{2}}=ak

These two results, c² = ap and b² = ak, are two of Euclid’s theorems which we shall study later.

If we now add together the two results using the addition property of equality we get :

\displaystyle \left. \begin{array}{l}{{c}^{2}}=ap\\{{b}^{2}}=ak\end{array} \right\rangle

\displaystyle {{c}^{2}}+{{b}^{2}}=ap+ak

\displaystyle \Rightarrow {{c}^{2}}+{{b}^{2}}=a\left( p+k \right)

\displaystyle \left( p+k \right)=a

\displaystyle {{c}^{2}}+{{b}^{2}}={{a}^{2}}

This result is very important in mathematics. Here is our diagram again showing what it means to write :

Pythagorean-theorem-2

We see that the area of the big square of side a is equal to the sum of the areas of the two small squares of sides b and c.

Now in a right – angled triangle we call the side opposite the right angle the hypotenuse. We call the other two sides the right sides. So a is the length of the hypotenuse and b and c are the lengths of the right sides.

We see that : in any right – angled triangle the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the two right sides.

This fact is know as Pythagoras’ theorem after the mathematician Pythagoras of about 500 B.C. It is said that the idea came to him while he was looking at a tile pattern of the kind shown in the diagram :

Pythagorean-theorem-3

We have shaded some sguares to show what Pythagoras noticed.

Pythagorean-theorem-4

The triangle is right – angled and we see that :

the area on the hypotenuse =5² =25 unit squares

the area on the 4 units right side = 4²  =16 unıt squares

the area on the 3 units right side = 3²  =9 unit squares

and

\displaystyle \begin{array}{l}25=16+9\\{{5}^{2}}={{4}^{2}}+{{3}^{2}}\end{array}

SUMMARY

Pythagorean-theorem-5

\displaystyle {{x}^{2}}={{y}^{2}}+{{z}^{2}}\Rightarrow x=\sqrt{{{y}^{2}}+{{z}^{2}}}

\displaystyle {{y}^{2}}={{x}^{2}}-{{z}^{2}}\Rightarrow y=\sqrt{{{x}^{2}}-{{z}^{2}}}

\displaystyle {{z}^{2}}={{x}^{2}}-{{y}^{2}}\Rightarrow z=\sqrt{{{x}^{2}}-{{y}^{2}}}





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