Question

Find missing part

Answer started below x=

Answer 1

a=3.75 un., b=11.25 un.

`x=7.5\ un.`

`y=(15√(3))/(4)\ un.`

`z=(15√(3))/(2)\ un.`

Explanation:

Given triangle is special 30°-60°-90° right triangle. The leg that is opposite to the angle of measure 30° is always equal to half of the hypotenuse. The hypotenuse is of length 15 units, the leg that is opposite to the 30° angle is leg with length of x units, then

`x=(15)/(2)=7.5\ un.`

In right triangle with hypotenuse x and legs y and a, angle opposite to the leg a is 30°, then

`a=(x)/(2)=(7.5)/(2)=3.75\ un.`

and

`b=15-a=15-3.75=11.25\ un.`

By the Pythagorean theorem,

`x^2=y^2+a^2,\\ \\7.5^2=3.75^2+y^2,\\ \\y^2=\left((15)/(2)\right)^2-\left((15)/(4)\right)^2=(225)/(4)-(225)/(16)=(675)/(16),\\ \\y=(15√(3))/(4)\ un.`

In right triangle with legs y and b and hypotenuse z, leg y is opposite to 30° angle, then

`z=2y=(15√(3))/(2)\ un.`

Answer 2

`x=7.5`

Explanation:

If you look at the large triangle, you will see that the side
`x` is opposite of 30° angle and the side 15 is opposite of 90° angle (side opposite of 90° angle is called the hypotenuse).

We have the angle, we have the side opposite and the hypotenuse. Which trigonometric ratio relates opposite side and hypotenuse? SINE!

So we use the sin ratio and cross multiply and solve for
`x`:

`Sin(A)=(Opposite)/(Hypotenuse)\\Sin(30)=(x)/(15)\\x=15*Sin(30)\\x=15*(0.5)\\x=7.5`

Hence,
`x=7.5`