Which of the following could be the ratio of the length of the longer leg of a 30-60-90 to the length of its hypotenuse? CHECK ALL THAT APPLY.A. √3 : 2B. 3 : 2√3C. 1 : √3D. 3√3 : 6E. √3 : √3F. √2 : √3-Apex Learning, Geometry

Answer: A. √3 : 2D. 3√3 : 6Step-by-step explanation:In a triangle described as 30°-60°-90° triangle, the base angles are 90° and 60°The side with angles 90° and 60° is the shortest leg and can be represented by 1 unitThe hypotenuse side is assigned a value twice the shorter leg value, which is 2 unitsFrom Pythagorean relationship; the square of the hypotenuse side subtract the square of the shorter leg gives the square of the longer sideThis is to say if;The given the shorter leg = 1 unitThe hypotenuse is twice the shorter leg= 2 units The longer leg is square-root of the difference between the square of the hypotenuse and that of the shorter leg [tex]=2^2-1^2\\\\=4-1=3\\\\\\b^2=3\\\\\\b=\sqrt{3}[/tex]where the longer leg is represented by side b in the Pythagorean theorem, the hypotenuse by c and the shorter leg by a to make;[tex]a^2+b^2=c^2[/tex]Hence the summary isa=shorter leg= 1 unitb=longer leg = √3 unitsc=hypotenuse=2 unitsThe ratio of longer leg to its hypotenuse is=√3:2⇒ answer option AThis is the same as 3√3:6 ⇒answer option D because you can divide both sides of the ratio expression by 3 and get option A[tex]=3\sqrt{3} :6\\\\\\\frac{3\sqrt{3} }{3} :\frac{6}{3} \\\\\\=\sqrt{3} :2[/tex]Answers are :option A and D