Question

In a right triangle, one acute angle is double the other . Prove that the hypotenuse is
double the smaller side.
(hint; extend the smaller side in equal length in the opposite direction.. new triangle is
an equilateral triangle.)

Answer 1

Here is your answer to the question.

Let the right angle triangle be ABC

Given,
In a right-angled triangle, one acute angle is double the other.

Let ∠BAC = x, therefore ∠ACB = 2x
Now,
in triangle ABC
➡️ ∠ABC + ∠BAC + ∠ACB = 180  { sum of all the angles in a triangle is 180 degree}

➡️90° + x + 2x = 180°    { triangle is right angled triangle}

➡️ 90 + 3x = 180

➡️ 3x = 180 - 90

➡️3x = 90

➡️ x = 90/3

➡️ x = 30
So, ∠BAC = x = 30 degree
and ∠ACB = 2x = 2 X 30 = 60 degree

Now,
In triangle ABC,
➡️ cos 2x = BC/AC
➡️ cos 60 = BC/AC

➡️ 1/2 = BC/AC

➡️ AC = 2*BC

${ac \: = 2bc}...proved$
Therefore... Hypotenuse is double the smaller side.
____________________________
This is the answer.
Hope it helps you.