Answer:
The value of x equals 5
The measurement of arc EF is 58°
The measurement of arc GH is 55°
Step-by-step explanation:
Let’s go over some essential concepts related to circles
- The total degrees in a circle is 360°
- A diameter splits a circle into two equal arcs, each measuring 180°
- An arc's measure corresponds to the central angle that subtends it
In circle D
∵ FD and GD serve as radii
- An angle formed at the center of a circle, with sides along the radii, is termed a central angle
∴ ∠FDG functions as a central angle corresponding to arc FG
- The measure of a central angle matches the measure of the arc it subtends
∴ m∠FDG = measurement of arc FG
∵ m∠FDG = 67°
∴ Measurement of arc FG = 67°
∵ EH acts as a diameter of circle D
- This divides the circle into two equal arcs
∴ The measurement of arc EH = 180°
∵ Arc EH includes arcs EF, FG, and GH
∴ m arc EH = m arc EF + m arc FG + m arc GH
∵ m arc EF = (10 x + 8)°
∵ m arc FG = 67°
∵ m arc GH = (11 x)°
- Plugging them into the equation above
∴ 180 = (10 x + 8) + 67 + 11 x
- Combine like terms on the right side
∴ 180 = 21 x + 75
- Deduct 75 from each side
∴ 105 = 21 x
- Divide both sides by 21
∴ 5 = x
The value of x is 5
∵ m arc EF = (10 x + 8)°
- Replace x with 5
∴ m arc EF = 10(5) + 8 = 50 + 8 = 58°
The measurement of arc EF is 58°
∵ m arc GH = (11 x)°
- Replace x with 5
∴ m arc GH = 11(5) = 55°
The measurement of arc GH is 55°
