Circles, Chords, and Arcs: Practice

All four measures add to 360 degrees.

The arc measures are 60, 80, 110, and 110 degrees. They are positive and total 360 degrees.

Subtract the major arc from a full turn to get the minor arc.

The minor arc and smaller central angle both measure 84 degrees.

Equal perpendicular center distances in one circle give equal chords.

Both lengths then equal 26 cm.

The equal-radius hypothesis permits equal minor arc measures.

Substitute into either arc expression.

The common minor arc measure is 55 degrees.

Half the chord is 8 units. The radius is the hypotenuse.

Use the nonnegative distance.

The half chord is the other leg of a right triangle.

The full chord is twice 6 cm, or 12 cm.

Align the centers and the first endpoint rays. Dilation by the radius ratio maps one circle and its selected endpoints onto the other.

The chord length doubles from 7 cm to 14 cm.

The center and midpoint are distinct points on the perpendicular bisector of AB.

Their line is perpendicular to AB, so angle OMB is a right angle.

Use perpendicular bisection before Pythagoras.

The full chord has length 48 units.

At a fixed radius, equal perpendicular distances give equal chord lengths.

The radius would matter for computing the common distance, but it is not needed for this equality transfer.

A chord needs two distinct endpoints. A radius and a half chord would have to satisfy

Zero separation cannot give distinct endpoints, so there is no genuine chord.

Take any point of the target circle. First apply the inverse dilation, then undo the translation.

The inverse dilation restores the original radius.

The inverse translation returns the center as well. Every target point therefore has a preimage on the original circle. Checking only selected radii, chords, or a drawing does not establish coverage of all target points.