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The 8 cm circle radius and the location of the circle X gives the following values;

(a) tan(<XAB) = -(r - 8)/(2•r + 4) = (8 - r)/8

Which gives;

  • r = (8 - 4)÷2 = 2

(b) <XAB is approximately 0.644 radians

(c) The area of the shaded region is approximately 3.107 cm²

Which method can be used to analyze the figure?

8 × (8 - r) = 0.5 × 16 × (8 + r) × sin(A)

(8 - r) = (8 + r) × sin(A)

sin(A) = (8 - r)÷(8 + r)

(8 - r)² = 8² + (8 + r)² - 2×8×(8 + r)×cos(A)

16×(8 + r)×cos(A) = (8² + (8 + r)²) - (8 - r)²

Which gives;

cos(A) = ((8² + (8 + r)²) - (8 - r)²) ÷ (16×(8 + r))

cos(A) = (2•r + 4)/(r + 8)

tan(A) = ((8 - r)÷(8 + r))/( (2•r + 4)/(r + 8))

tan(A) = -(r - 8)/(2•r + 4) = (8 - r)/8

(8 - r)/(2•r + 4) = (8 - r)/8

8 = 2•r + 4

2•r + 4 = 8

Therefore;

  • r = (8 - 4)÷2 = 2

(b) sin(A) = (8 - r)÷(8 + r)

sin(A) = (8 - 2)÷(8 + 2) = 0.6

Therefore;

<XAB = <A = arcsin(0.6) ≈ 0.644 rad

<XAB in degrees ≈ 36.87°

Angle at sector PXQ = 180° - 2 × (36.87°) ≈ 106.26°

Area of sector PXQ ≈ (106.26°/(360°)) × π × 2²

(106.26°/(360°)) × π × 2² ≈ 3.71

Area of sector APO ≈ (36.87°/(360°)) × π × 8² ≈ 20.59

Area of triangle AXB = 8 × (8-2) = 48

The shaded area is therefore;

48 - (2×20.59 + 3.71) ≈ 3.107

  • The area of the shaded region, A ≈ 3.107

Learn more about the trigonometric ratios here:

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a) The radius of the small circle is 2 centimeters.

b) The measure of the angle XAB is approximately equal to 51.340°.

c) The area of the shaded region is approximately equal to 2.423 square centimeters.

How to analyze a system formed by three semicircles and a circle

In this question we must analyze a geometrical system formed by three semicircles and a circle. The small circle touches the two side semicircles tangentially at points P and Q and the uppermost section of it also tangent to the central semicircle (point R).

(a) Therefore, we have to solve the following system of equations:

8² + h² = (8 + r)²      (1)

h + r = 8      (2)

By (2) in (1):

8² + (8 - r)² = (8 + r)²

32 · r = 64

r = 2

The radius of the small circle is 2 centimeters.

(b) The measure of the angle XAB is found by trigonometric functions:

tan m ∠ XAB = OX / AO

tan m ∠ XAB = 10 / 8

m ∠ XAB ≈ 51.340°

The measure of the angle XAB is approximately equal to 51.340°.

(c) The area of the shaded region if the area of the triangle AXB minus the areas of the three circular sections, that is:

A = 0.5 · (16 cm) · (10 cm) · sin 51.340° - (51.340° / 180°) · π · (8 cm)² - 0.5 · (77.320° / 180°) · π · (2 cm)²

A ≈ 2.423 cm²

The area of the shaded region is approximately equal to 2.423 square centimeters.

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