AB is the common chord of O and Q. If AB = 10 and each circle has a radius of length 13, how long is OQ

Question

Two circles are side by side and partially overlap. The circle on the left has center point O. The circle on the right has center point Q. There are 2 points and 2 line segments.

• The right side of ⊙O and the left side of ⊙Q intersect twice: the upper point of intersection is A, and the lower point of intersection is B.
• The first segment begins at O, goes to the right, intersects the left side of ⊙Q, intersects the right side of ⊙O, and ends at Q.
• The second segment begins at A, goes down, intersects segment O Q, and ends at B.

Answer 1

Join A to O and Q, and B to O and Q. AQ=AO=BQ=BO=13 because they are all radii. Figure AQBO is a rhombus with diagonals AB and OQ bisecting one another perpendicularly. Call their intersection point N. AN=NB=5 because AN=AN+NB=10.

In right ∆ANO, AO²=AN²+ON² so:

13²=5²+ON², ON=√(13²-5²)=√(169-25)=√144=12.

ON=NQ=12, and OQ=ON+NQ=24.