Question

when the length of the hypotenuse of a triangle is equal to the length of one of the legs times \\(\sqrt{2}\\), it is a \\(30^\circ - 60^\circ - 90^\circ\\) triangle.
\\(\bigcirc\\) true
\\(\bigcirc\\) false

Explanation:

Step1: Recall triangle properties

In a \(30^\circ - 60^\circ - 90^\circ\) triangle, the ratios of the sides are \(1:\sqrt{3}:2\) (opposite \(30^\circ\), \(60^\circ\), \(90^\circ\) respectively). In a \(45^\circ - 45^\circ - 90^\circ\) triangle, the ratios are \(1:1:\sqrt{2}\) (legs:leg:hypotenuse).

Step2: Analyze the given condition

If hypotenuse \(= \text{leg} \times \sqrt{2}\), let leg length be \(x\), then hypotenuse is \(x\sqrt{2}\). By Pythagorean theorem, the other leg is also \(x\) (since \(x^2 + x^2=(x\sqrt{2})^2\)), so it's a \(45^\circ - 45^\circ - 90^\circ\) triangle, not \(30^\circ - 60^\circ - 90^\circ\).

Answer:

False