the moon forms a right triangle with the earth and the sun during one of its phases, as shown below:
a scientist measures the angle x and the distance y between the sun and the moon. using complete sentences, explain how the scientist can
use only these two measurements to calculate the distance between the earth and the moon.

The distance between the Earth and the Moon is equal to the distance between the Sun and the Moon multiplied by the sine of angle x

Step-by-step explanation:

Let

EM > the distance between the Earth and the Moon.

y > the distance between the Sun and the Moon.

we know that

In the right triangle of the figure

The sine of angle x is equal to divide the opposite side to angle x (distance between the Earth and the Moon.) by the hypotenuse ( distance between the Sun and the Moon)

so

sin(x)=EM/y

Solve for EM

EM=(y)sin(x)

therefore

The distance between the Earth and the Moon is equal to the distance between the Sun and the Moon multiplied by the sine of angle x

center (1,3) with radius 6

step-by-step explanation:

we must complete the square to find the center and radius of the circle.

first make sure the x and y squared terms have 1 as their coefficients. we also make sure x and y terms together.

we now create space between the x and y terms with parenthesis.

we complete the square by taking the middle terms -2x and the -6y - divide each and square them.

we add the squares to both sides.

simplify.

and write the quadratics in factored form.

the center is (h,k) or (1,3). the radius is the square root of 36 which is 6.