Question

ABC is a right angled triangle and right angled at B such that angle BCA= 2 angle BCA. Show that hypotenuse AC=2BC​

Answer 1

In △ABD and △ABC we have BD=BC

AB=AB [Common]

∠ABD=∠ABC=90

∘

∴ By SAS criterion of congruence we get

△ABD≅△ABC

⇒AD=AC and ∠DAB=∠CAB [By CPCT]

⇒AD=AC and ∠DAB=x [∴∠CAB=x]

Now, ∠DAC=∠DAB+∠CAB=x+x=2x

∴∠DAC=∠ACD

⇒DC=AD [Side Opposite to equal angles]

⇒2BC=AD since DC=2BC

⇒2BC=AC Since AD=AC

Hence proved.