Question

Determine the value of x using the information in the diagram.

Answer 1

Given:

Let's denote the intersection of the two chords as O.

Given that the measure of arc CD is 99 degrees,

The measure of angle COD = 72 degrees.

We need to find the measure of arc AB.

Measure of arc AB:

We can apply the property stating that "if two chords intersect at the interior of the circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle."

Using this property, we have:

`\[ \text{angle COD} = (1)/(2) (\text{measure of arc CD} + \text{measure of arc AB}) \]`

Substituting angle COD = 72 degrees, measure of arc CD = 99 degrees, and measure of arc AB = x, we get:

`\[ 72 \text{ degrees} = (1)/(2) (99 \text{ degrees} + x) \]`

Multiplying both sides of the equation by 2, we have:

`\[ 144 \text{ degrees} = 99 \text{ degrees} + x \]`

Subtracting both sides of the equation by 99, we get:

`\[ 45 \text{ degrees} = x \]`

Thus, the value of x is 45 degrees.

Hence, Option B is the correct answer.

Answer 2

Given:

Let us denote the intersection of the two chords be O.

Given that the measure of arc CD is 99°

The measure of ∠COD = 72°

We need to determine the measure of arc AB

Measure of arc AB:

Let us use the property that, "if two chords intersect at the interior of the circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle".

Using the above property, we have;

`\angle COD=(1)/(2)(m \widehat{CD} + m \widehat{AB})`

Substituting ∠COD = 72° ,
`m \widehat {CD}= 99^(\circ)`,
`m \widehat {AB} = x`, we get;

`72^(\circ)=(1)/(2)(99^(\circ)+x)`

Multiplying both sides of the equation by 2, we have;

`144^(\circ)=99^(\circ)+x`

Subtracting both sides of the equation by 99, we get;

`45^(\circ)=x`

Thus, the value of x is 45°

Hence, Option B is the correct answer.