Question

Find x please help thank you A. 33–√ B. 32√2 C. 3 D. 6–√

Answer 1

3/√2 or 3√2/2

Explanation:

Bigger triangle has internal angles 45/45/90, so the legs have equal length and = 6/√2 as hypotenuse = 6

Smaller triangle has hypotenuse of 6/√2 and the angle against leg x is 30°

As we know the leg opposite to 30° angle is half the hypotenuse, so

• x= 1/2*6/√2= 3/√2= 3√2/2

Answer 2

`\textbf{B. }\ (3)/(2)√(2)`

Explanation:

You know the short side of an isosceles right triangle is 1/√2 times the hypotenuse. So, the hypotenuse of the 30°-60°-90° right triangle is ...

6/√2 = (6√2)/2 = 3√2

You also know the short leg of a 30°-60°-90° right triangle is half the length of its hypotenuse, so ...

x = (1/2)(3√2)

x = (3/2)√2

_____

Additional comment

For many trig and geometry problems, it is useful to remember the ratios of side lengths for "special" right triangles:

isosceles right triangle: 1 : 1 : √2

30°-60°-90° triangle: 1 : √3 : 2 (shortest to longest)

This knowledge usefully relates to the trig function values for these "special" triangles.