Question

eh is a diameter of ⊙d. the measure of $overset{\frown}{ef}$ is $(10x + 8)^{circ}$ and the measure of $overset{\frown}{gh}$ is $(11x)^{circ}$. determine the values. the value of x is. the measure of $overset{\frown}{ef}$ is degrees. the measure of $overset{\frown}{gh}$ is degrees.

Explanation:

Step1: Recall circle - arc property

The sum of the measures of arcs that form a semi - circle is 180 degrees. Since EH is a diameter, the sum of the measures of arc EF, the 67 - degree arc, and arc GH is 180 degrees. So, $(10x + 8)+67+(11x)=180$.

Step2: Combine like terms

Combine the x - terms and the constant terms: $10x+11x + 8+67=180$, which simplifies to $21x+75 = 180$.

Step3: Isolate the variable term

Subtract 75 from both sides of the equation: $21x=180 - 75$, so $21x=105$.

Step4: Solve for x

Divide both sides by 21: $x=\frac{105}{21}=5$.

Step5: Find the measure of arc EF

Substitute $x = 5$ into the expression for the measure of arc EF: $10x+8=10\times5 + 8=50 + 8=58$ degrees.

Step6: Find the measure of arc GH

Substitute $x = 5$ into the expression for the measure of arc GH: $11x=11\times5 = 55$ degrees.

Answer:

The value of $x$ is 5.
The measure of $\widehat{EF}$ is 58 degrees.
The measure of $\widehat{GH}$ is 55 degrees.