Question

In the given figure ABC is a right triangle and right angled at B such ∠BCA=2∠BAC.
Show that hypotenuse AC=2BC.
(Hint: Produce CB to a point D that BC=BD)​

Answer 1

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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.