Question

Prove the statement given”In a right angle triangle ,the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Answer 1

Hint: Consider a right angle triangle whose vertices are denoted A,B,C respectively and AB, BC, CA are the sides of triangle respectively.And angle ABC should be equal to 90 degrees that is $\left| \!{\underline {\, ABC \,}} \right. $=${{90}^{0}}$. An AC is a hypotenuse whose square is equal to the sum of the squares of the other two sides.

Complete step by step answer:
This theorem is Pythagoras Theorem which is also called as Pythagorean Theorem.
Statement: Pythagorus theorem states that “In a right - angled triangle, the square of the hypotenuse side is equal to the sum of the squares of the other two sides ”.The sides of this triangle have been named as Perpendicular,Base and Hypotenuse.Here the hypotenuse is the longest side, as it is opposite to angle .The sides of a right a right angle (say a,b,c) which have positive integer values ,when squared ,are put into an equation ,also called a Pythagorean triple.
Proof: To prove $A{{C}^{2}}=A{{B}^{2}}+B{{C}^{2}}$
Construction:

Let us Consider a $\Delta ABC$ and draw BD ⊥ AC
{∵If a perpendicular is drawn from the vertex of a right angle triangle to the hypotenuse then triangle
  triangle on both sides of the perpendicular are similar to the whole triangle and to each other.}
 ∴$\Delta ADB\sim \Delta ABC$
 Since the sides of similar triangles are in the same ratio,
$\Rightarrow \dfrac{AD}{AB}=\dfrac{AB}{AC}$
$\Rightarrow AD.AC=A{{B}^{2}}$⟶equation(1)

Similarly,
$\Delta BDC\sim \Delta ABC$
Since the sides of similar triangles are in the same ratio,
$\Rightarrow \dfrac{CD}{BC}=\dfrac{BC}{AC}$
  ⇒$CD.AC=B{{C}^{2}}$⟶equation(2)
On adding equation(1)&(2) we get
$\begin{align}
  & AD.AC+CD.AC=A{{B}^{2}}+B{{C}^{2}} \\
 & AC(AD+CD)=A{{B}^{2}}+B{{C}^{2}} \\
 & AC\times AC=A{{B}^{2}}+B{{C}^{2}} \\
 & A{{C}^{2}}=A{{B}^{2}}+B{{C}^{2}} \\
\end{align}$
Therefore, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Hence proved..

Note:
Here this theorem is only applicable for right - angle triangle whose one of the vertex angles is And we should draw the perpendicular to the hypotenuse which will make it easy to prove the theorem. And also the values of sides must be positive.