In a right-angled trapezoid, the middle line is the average of the lengths of the two parallel sides. Let's call the two parallel sides of the trapezoid "a" and "b", and the length of the middle line "m".
Since the radius of the circle inscribed in the trapezoid is 4 cm, we know that the distance from the center of the circle to each side of the trapezoid is 4 cm. Let's call this distance "r".
We can draw a perpendicular from the center of the circle to one of the parallel sides of the trapezoid, as shown in the diagram below:
_______
/ \
/ \
/ r \
/____________\
a
Since the acute angle of the trapezoid is 30°, we know that the angle between the perpendicular and the shorter parallel side is 60° (since the two angles add up to 90°). Therefore, we can use trigonometry to find the length of the shorter parallel side:
tan(60°) = r/a
a = r/tan(60°)
a = 4/tan(60°)
a ≈ 4*1.732 ≈ 6.928
Similarly, we can draw a perpendicular from the center of the circle to the longer parallel side of the trapezoid, as shown in the diagram below:
_______
/ \
/ \
/ r \
/____________\
b
Since the two perpendiculars and the longer parallel side form a right triangle, we can use the Pythagorean theorem to find the length of the
b² = (2r)² - m²
But we don't know the length of the middle line yet. However, we can use the fact that the middle line is the average of the lengths of the two parallel sides:
m = (a + b)/2
b = 2m - a
Substituting this expression for b into the equation above, we get:
(2m - a)² = (2r)² - m²
4m² - 4am + a² = 4r² - m²
5m² - 4am + a² = 4r²
5m² - 4(4/tan(60°))m + (4²) = 4(4²)
5m² - 16(√3)m + 16 = 64
5m² - 16(√3)m - 48 = 0
We can solve this quadratic equation using the quadratic formula:
m = (-b ± sqrt(b² - 4ac))/2a
In this case, a = 5, b = -16(√3), and c = -48. Substituting these values into the quadratic formula, we get:
m = (-(-16(√3)) ± sqrt((-16(√3))² - 4(5)(-48)))/(2(5))
m = (16(√3) ± sqrt(256(3) + 960))/10
m = (16(√3) ± sqrt(1728))/10
m = (16(√3) ± 24)/10
We take the positive root since the length of the middle line must be positive. Therefore,
m = (16(√3) + 24)/10
m = (8(√3) + 12)/5
m ≈ 5.797
Now we can use this value of m to find the length of the longer parallel side:
b = 2m - a
b = 2(5.797) - 6.928
b ≈ 4.666
Therefore, the shorter parallel side has length a ≈ 6.928 cm and the longer parallel side has length b ≈ 4.666 cm.