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 : 2
b. 3 : 2√3
c. 1 : √3
d. 3√3 : 6
e. √3 : √3
f. √2 : √3
-apex learning, geometry

A. √3 : 2

D. 3√3 : 6

Step-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 unit

The hypotenuse side is assigned a value twice the shorter leg value, which is 2 units

From Pythagorean relationship; the square of the hypotenuse side subtract the square of the shorter leg gives the square of the longer side

This is to say if;

The given the shorter leg = 1 unit

The 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

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;

Hence the summary is

a=shorter leg= 1 unit

b=longer leg = √3 units

c=hypotenuse=2 units

The ratio of longer leg to its hypotenuse is

=√3:2⇒ answer option A

This 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

Answers are :option A and D

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