The image contains a diagram of two identical circles intersecting each other, with centers $A$ and $B$ respectively. Points $A, B, C,$ and $D$ lie on the circumference of the circles. The task is to find the degree of the angle subtended by chord $CD$ at the center of circle $B$.

Analysis

1. Understanding the Diagram:

   • The two circles intersect at points $C$ and $D$.
   • Center $A$ belongs to one circle, and center $B$ belongs to the other.
   • The line segment $CD$ passes through the points where the circles intersect.
   • The question asks for the angle subtended by chord $CD$ at the center of circle $B$.

2. Key Observations:

   • Since the circles are identical, they have the same radius, say $r$.
   • The line segment $AB$ connects the centers of the two circles.
   • $CD$ is a common chord for both circles and is perpendicular to $AB$.

3. Geometric Properties:

   • Since $CD$ is a common chord and the circles are identical, $AB$ bisects $CD$ at right angles.
   • Triangle $BCD$ is isosceles with $BC = BD = r$ (radius of circle with center $B$).

4. Finding the Angle $\angle CBD$:

   • In the isosceles triangle $BCD$, $\angle CBD = 90^\circ$ because the line $AB$ is perpendicular to the chord $CD$.

Thus, the degree of the angle subtended by chord $CD$ at the center of circle $B$ is 90 degrees.

Would you like more details or explanations?

Related Questions:

1. How do you calculate the length of a chord in a circle?
2. How can we prove that the angle subtended by a diameter of a circle is 90 degrees?
3. What is the relationship between the radius of a circle and its chords?
4. How do you find the area of a segment of a circle?
5. How do intersecting chords within a circle relate to each other?

Tip:

When solving problems involving intersecting circles, look for symmetry and consider the properties of common chords and perpendicular bisectors.