State whether the statement is true or false.

If in a right-angled triangle, one acute angle is double the other then the hypotenuse is double the smallest side.

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True

False

answer is A.

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Detailed Solution

Given, in a right-angled triangle, one acute angle is double the other.
Draw a triangle according to the given statement.
Draw another triangle similar to triangle ABC.

We know that the sum of angles in a triangle is 180°.
Hence, 90° + x° + 2x° = 180°.

\Rightarrow

3x + 90° = 180°

\Rightarrow

3x = 90°

\Rightarrow

x = 30°
So, 2x = 2(30).

\Rightarrow

2x = 60°
So, ∠C = 60°.
Similarly, ∠A = 60° and ∠B = 60°.
So, we can say that

Δ

ADC is an equilateral triangle.
Hence, AD = AC = DC.
Consider AC = DC.

\Rightarrow

AC = DB + BC

\Rightarrow

AC = BC + BC [since, the two triangles are similar]

\Rightarrow

AC = 2BC
AC is a hypotenuse and BC is the small side.
So, we can say that the hypotenuse is double the smallest side.
Hence, the given statement is true and option 1 is correct.

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