Question

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

Answer 1

A. √3 : 2

D. 3√3 : 6

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

`=2^2-1^2\\\\=4-1=3\\\\\\b^2=3\\\\\\b=√(3)`

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;

`a^2+b^2=c^2`

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

`=3√(3) :6\\\\\\(3√(3) )/(3) :(6)/(3) \\\\\\=√(3) :2`

Answers are :option A and D