The moon forms a right triangle with the earth and the sun during one of its phases, as shown below: a right angle triangle is shown with the earth at the right angle. the acute angle between the line joining the earth and the s 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.

As per your description these are the data:

1) right triangle

2) distance between the Moon and the Sun = hypotenuse

3) distance between Earth and the Sun = opposite - leg to angle x: call it S

So, you can use the trigonometric function tangent, which is tan (x) = opposite-leg to hypotenuse, or tan (x) = S / y.

Therefore, you can solve for S and find S = y tan(x).

That is the distance between the Sun and the Earth is equal to the tangent of the angle x times the distance between the Sun and the Moon.

If I read correctly, x is the acute angle that the sun makes with the lines connecting Earth and Moon; y is the line connecting the sun with the moon, aka the distance between.
Since this is a right triangle, we can use Trigonometry to solve the sides and angles.
For example, if we know any two features, 2 sides, 2 angles, or a side and an angle, we can calculate the others.
So, if we say that the angle x = 50°, then the value of y can be found by:
SOH... Sin50 = O/H = opposite/hypotenuse
If we know then distance from Earth to Moon, call it "o," and the hypotenuse = y, then:
Sin50 = o/y
y = o/sin50.
Alternatively, if we know x, say 50° again, then we can find the top angle at the apex (formed from moon to Earth and moon to.sun), call it "a," because the sum of 3 angles of a triangle always = 180°.
a = 180 - (90+50) = 180-140 = 40°
Additionally, if we already know the distance from Earth to moon, or height of triangle (call it "h"), then we can use law of Sines to find the distance from Earth to Sun (d):
(a/sinA) = (b/sinB)
(d/sin40) = (h/sin50)
Let's say that distance h = 500km, then:
d = (hsin40)/sin50 = 500sin40/sin50 = 500(0.643)/
= 321.4/0.766 = 419.55 km

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