Question

EH is a diameter of CircleD. The measure of Arc E F is (10x + 8)° and the measure of Arc G H is (11x)°. Circle D is shown. Line segment E H is a diameters. Lines segments F D and G D are radii. Angle F D G is 67 degrees. Determine the values. The value of x is . The measure of Arc E F is degrees. The measure of Arc G H is degrees.

Answer 1

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°

Answer 2

Answer:

The value of x is

5

.

The measure of Arc E F is

58

degrees.

The measure of Arc G H is

55

degrees.