Find the measure of arc DB.

Question

Circle A is intersected by line CD at points D and E and line CB at point B, forming angle ECB outside of the circle, the measure of angle ECB is 5x degrees, arc EB is 96 degrees, and arc DB 25x plus 21 degrees.

25°
96°
118°
146°

Answer 1

To find the measure of arc DB, we can use the properties of angles and arcs in circles.

Given:

• The measure of angle \( \angle ECB = 5x \) degrees
• The measure of arc \( EB = 96 \) degrees
• The measure of arc \( DB = 25x + 21 \) degrees

According to the properties of circles, the angle formed outside the circle by two secant lines is equal to one-half the difference of the measures of the arcs intercepted by those secant lines. Thus, we have the following equation:

\[ \angle ECB = \frac{1}{2} (\text{measure of arc DB} - \text{measure of arc EB}) \]

Substituting the known values:

\[ 5x = \frac{1}{2} ((25x + 21) - 96) \]

Now, simplify the right side:

\[ 5x = \frac{1}{2} (25x + 21 - 96) \] \[ 5x = \frac{1}{2} (25x - 75) \] \[ 5x = \frac{25x - 75}{2} \]

To eliminate the fraction, multiply both sides by 2:

\[ 10x = 25x - 75 \]

Now, solve for \( x \):

\[ 10x - 25x = -75 \] \[ -15x = -75 \] \[ x = 5 \]

Now substitute \( x = 5 \) back to find the measure of arc DB:

\[ \text{measure of arc DB} = 25x + 21 = 25(5) + 21 = 125 + 21 = 146 \text{ degrees} \]

Therefore, the measure of arc DB is:

\[ \boxed{146} \]