Circles \( C, P \), and \( Q \) have radii 1,2 , and 3 respectively. If the three circles are tangent to each other, what is the nature of the triangle formed by joining the three centers of the circles? A. Isosceles B. Right angled C. Right isosceles D. Equilateral E. Cannot be determined with the information given

To determine the nature of the triangle formed by the centers of the circles \( C, P, \) and \( Q \), we first need to find the distances between the centers of the circles based on their radii. Let the centers of circles \( C, P, \) and \( Q \) be denoted as \( O_C, O_P, \) and \( O_Q \) respectively. The radii of the circles are given as follows: - Circle \( C \) has radius \( r_C = 1 \) - Circle \( P \) has radius \( r_P = 2 \) - Circle \( Q \) has radius \( r_Q = 3 \) Since the circles are tangent to each other, the distance between the centers of any two circles is equal to the sum of their radii. 1. **Distance between \( O_C \) and \( O_P \)**: \[ d_{CP} = r_C + r_P = 1 + 2 = 3 \] 2. **Distance between \( O_P \) and \( O_Q \)**: \[ d_{PQ} = r_P + r_Q = 2 + 3 = 5 \] 3. **Distance between \( O_C \) and \( O_Q \)**: \[ d_{CQ} = r_C + r_Q = 1 + 3 = 4 \] Now we have the lengths of the sides of the triangle formed by the centers: - \( O_C O_P = 3 \) - \( O_P O_Q = 5 \) - \( O_C O_Q = 4 \) Next, we can check the nature of the triangle using the triangle inequality and the Pythagorean theorem. To check if the triangle is a right triangle, we can see if the squares of the lengths satisfy the Pythagorean theorem: \[ a^2 + b^2 = c^2 \] where \( c \) is the longest side. Here, the sides are: - \( a = 3 \) - \( b = 4 \) - \( c = 5 \) Calculating the squares: \[ 3^2 + 4^2 = 9 + 16 = 25 \] \[ 5^2 = 25 \] Since \( 3^2 + 4^2 = 5^2 \), the triangle formed by the centers of the circles is a right triangle. Thus, the answer is: \[ \boxed{\text{B. Right angled}} \]