Question

calculating arc measures
eh is a diameter of
odot d. the measure of overarc{ef} is (10x + 8)^{circ} and the
measure of overarc{gh} is (11x)^{circ}.
determine the values.
the value of x is
the measure of overarc{ef} is degrees.
the measure of overarc{gh} is degrees.

Explanation:

Step1: Use the property of a semicircle

Since \(EH\) is a diameter, the sum of the measures of \(\overset{\frown}{EF}\), \(\angle FDG\), and \(\overset{\frown}{GH}\) is \(180^{\circ}\). So, \((10x + 8)+67+11x=180\).

Step2: Simplify the equation

Combine like - terms: \(10x+11x+8 + 67=180\), which gives \(21x+75 = 180\).

Step3: Solve for \(x\)

Subtract \(75\) from both sides: \(21x=180 - 75\), so \(21x=105\). Then divide both sides by \(21\): \(x=\frac{105}{21}=5\).

Step4: Find the measure of \(\overset{\frown}{EF}\)

Substitute \(x = 5\) into \(10x+8\). So, \(10\times5+8=50 + 8=58^{\circ}\).

Step5: Find the measure of \(\overset{\frown}{GH}\)

Substitute \(x = 5\) into \(11x\). So, \(11\times5 = 55^{\circ}\).

Answer:

The value of \(x\) is \(5\).
The measure of \(\overset{\frown}{EF}\) is \(58\) degrees.
The measure of \(\overset{\frown}{GH}\) is \(55\) degrees.