Question

Anjeli writes the equation (a+b)2=c2+4(12ab) to begin a proof of the Pythagorean theorem.

Answer 1

Answer:

Therefore Anjeli is correct and

$(a+b)^{2}=c^{2}+4(\frac{1}{2}ab)$  is TRUE.

Step-by-step explanation:

Given:

Label on triangle as ΔABC right angle at C such that

AB = c  .....(Hypotenuse)

AC = b  ....(Longer leg)

BC = a  ....(Shorter leg)

To Prove:

$(a+b)^{2}=c^{2}+4(\frac{1}{2}ab)$

Proof:

We know, in Right angle triangle ABC by Pythagoras theorem we get,

$(\textrm{Hypotenuse})^{2} = (\textrm{Shorter leg})^{2}+(\textrm{Longer leg})^{2}$

Substituting the values we get

$AB^{2}=BC^{2}+AC^{2}\\\\c^{2}=a^{2}+b^{2}$    ...............( 1 )

Now the Left hand side of what Anjeli wrote is

Left hand side = (a+b)²

 Using identity (A+B)² = A²+ B² + 2AB we get

Left hand side = a²+ b² + 2ab

From ( 1 ) we have $c^{2}=a^{2}+b^{2}$

Substituting we get

Left hand side = c² + 2ab        ...........................( 2 )

Now,

Right hand side =$c^{2}+4(\frac{1}{2}ab)$

Dividing 4 by 2 we get  2, hence

Right hand side =$c^{2}+2ab$  .........................( 3 )

Therefore,

Left hand side = Right hand side          From  ( 2 ) and ( 3 )

$(a+b)^{2}=c^{2}+4(\frac{1}{2}ab)$      ..True

Therefore Anjeli is correct and

$(a+b)^{2}=c^{2}+4(\frac{1}{2}ab)$  is TRUE.

Answer 2

Answer:a+b  c   1/2ab

Step-by-step explanation: