Answer:
The value of x is 5
The measure of arc EF is 58°
The measure of arc GH is 55°
Step-by-step explanation:
Let us revise some important facts in a circle
- The measure of a circle is 360°
- Any diameter divides the circle into two equal arcs the measure of each one is 180°
- The measure of an arc equal the measure of the central angle subtended by it
In circle D
∵ FD and GD are radii
- The angle whose vertex is the center of a circle and its sides
are radii in the circle is called a central angle
∴ ∠FDG is a central angle subtended by arc FG
- The measure of the central angle is equal to the measure
of its subtended arc
∴ m∠FDG = measure of arc FG
∵ m∠FDG = 67°
∴ Measure of arc FG = 67°
∵ EH is a diameter of circle D
- It divides the circle into two equal arcs
∴ The measure of arc EH = 180°
∵ Arc EH contains 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)°
- Substitute them in the equation above
∴ 180 = (10 x + 8) + 67 + 11 x
- Add the like terms in the right hand side
∴ 180 = 21 x + 75
- Subtract 75 from both sides
∴ 105 = 21 x
- Divide both sides by 21
∴ 5 = x
The value of x is 5
∵ m arc EF = (10 x + 8)°
- Substitute x by 5
∴ m arc EF = 10(5) + 8 = 50 + 8 = 58°
The measure of arc EF is 58°
∵ m arc GH = (11 x)°
- Substitute x by 5
∴ m arc GH = 11(5) = 55°
The measure of arc GH is 55°
