Question

7. ABC is an isosceles triangle with AC = BC. If AB2 = 2AC2 prove that ABC is a right triangle​

Answer 1

Diagram:

$\setlength{\unitlength}{1 cm}\begin{picture}(0,0)\thicklines\qbezier(1, 0)(1,0)(3,3)\qbezier(5,0)(5,0)(3,3)\qbezier(5,0)(1,0)(1,0)\put(2.85,3.2){ \bf A }\put(0.5,-0.3){ \bf C }\put(5.2,-0.3){ \bf B }\end{picture}$

Given:

• ΔABC is an isosceles Δ
• AC = BC
• AB² = 2 AC²

To Prove:

• ΔABC is a Right Angled Triangle

Proof:

We know that

AB² = 2 AC²

⇒ AB² = AC² + AC²

Since AC = BC,

AB² = AC² + BC²

Now we know that AB is the longest side or Hypotenuse

Now let's check whether Pythagoras theorem holds good or not

(Hypotenuse)² = (Base)² + (Height)²

⇒ AB² = BC² + AC²

⇒ AB² = AC² + AC²   (BC = AC)

⇒ AB² = 2 AC²    

⇒ AB² = AB²            (Substituting given data)

LHS = RHS

So Pythagoras theorem has been Satisfied.

By Converse of Pythagoras theorem,

ΔABC is a Right Angled Triangle