How do I find these values in a triangle?

Question

I have a 90-60-30 triangle. The legs are 10 square root 3 and x. The hypotenuse is y. How do I find x and y?

Answer 1

Here AB = X, AC = 10√3 and BC = Y

Tan 60 = AB/AC

Trigonometric table in tan60 = √3 and sin60 = √3/2

√3 = x / 10√3

Multiply each side by 10√3.

√3(10√3) = x

x = 10(√3)²

x = 30

 

And sin 60 = AB/BC

√3/2 = x / y

√3/2 = 30 / y

Multiply each side by 'y'.

(√3/2)y = 30

Multiply each side by '2'.

√3 y = 60

Divide each side by √3.

y = 60 /√3.

There fore x = 30 and y = 60/√3

Comments

x =10 and y = 20

Answer 2

The side opposite to right angle is hypotenuse y.

The side opposite to angle A is hypotenuse 10√3.

The side opposite to angle C is hypotenuse x.

tan 30^o = opposite side / adjacent side = x  / 10√3

1/√3 = x  / 10√3

x = √3 / 10√3 = 10

By Pythagoras theorem,

Hypotenuse² = opposite side² + adjacent side²

y² = x² + (10√3)²

y² = 10² + (10√3)²

y² = 100 + 300

y = √400

y = 20

Therefore x = 10 and y = 20.