Question

18)
2√6
(right - angled isosceles triangle with one angle 45°, hypotenuse 2√6, legs y and x)

Explanation:

Step1: Identify the triangle type

This is a right - isosceles triangle (\(45^{\circ}-45^{\circ}-90^{\circ}\) triangle). In a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle, the ratio of the sides is \(x:x:x\sqrt{2}\), where the hypotenuse \(c=x\sqrt{2}\) and the legs are of equal length (\(a = b=x\)).
Given the hypotenuse \(c = 2\sqrt{6}\).

Step2: Use the side - ratio formula

We know that \(c=x\sqrt{2}\). Substituting \(c = 2\sqrt{6}\) into the formula \(x\sqrt{2}=2\sqrt{6}\).
Solve for \(x\):

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Since \(x = y\) (legs of a right - isosceles triangle are equal).

Answer:

\(x = 2\sqrt{3}\), \(y = 2\sqrt{3}\)