Question

geometry
examples #9 - 10, find the value of the variable.
9)
(5x - 13)°
106°
10)
3x°
(2x + 80)°
ch 10 notes: circles

Explanation:

Example 9

Step1: Set up the equation

In a circle, the measure of an arc is equal to the measure of its central angle. Since the two arcs are equal (assuming they are congruent arcs), we have \(5x - 13=106\).

Step2: Solve for \(x\)

Add \(13\) to both sides of the equation: \(5x-13 + 13=106+13\), which simplifies to \(5x = 119\). Then divide both sides by \(5\): \(x=\frac{119}{5}=23.8\).

Example 10

Step1: Use the property of a circle's total central - angle sum

The sum of central angles in a circle is \(360^{\circ}\). One of the angles is a straight - angle (\(180^{\circ}\)), so \(3x+(2x + 80)=180\).

Step2: Simplify the left - hand side

Combine like terms: \(3x+2x+80=(3 + 2)x+80=5x + 80\). So the equation becomes \(5x+80 = 180\).

Step3: Solve for \(x\)

Subtract \(80\) from both sides: \(5x+80-80=180 - 80\), which gives \(5x=100\). Then divide both sides by \(5\): \(x = 20\).

Answer:

• For Example 9: \(x = 23.8\)
• For Example 10: \(x = 20\)