These 24 problems practice circle theorems for Geometry, aligned to G-C.A.1 and G-C.A.2. They start with similar circles, central angles and inscribed angles, then move to semicircles, tangents and chords.
Later problems use inscribed quadrilaterals and algebra to find unknown angles and arcs. The last six are word problems and multi-step tasks set in plazas, ponds and amusement parks.
Problem 1: Similar circles
Circle A has radius 4 cm and circle B has radius 10 cm. (a) Describe a translation and a dilation that map circle A onto circle B and give the scale factor. (b) Find the ratio of the circumference of B to the circumference of A.
Problem 2: Ratio of circumferences
Circle P has radius 3 inches and circle Q has radius 12 inches. Find both circumferences in terms of \(\pi\) and the ratio of the circumference of Q to that of P.
Problem 3: Central angle and arc
In circle \(O\), central angle \(\angle AOB\) measures \(72^{\circ}\). Find the measure of minor arc \(AB\) and of major arc \(AB\).
Problem 4: Inscribed angle from an arc
Inscribed angle \(\angle ACB\) intercepts arc \(AB\) that measures \(110^{\circ}\). Find \(m\angle ACB\).
Problem 5: Arc from an inscribed angle
An inscribed angle measures \(38^{\circ}\). Find its intercepted arc and the central angle that intercepts the same arc.
Problem 6: Inscribed angle algebra
Inscribed angle \(\angle ABC\) measures \((3x+5)^{\circ}\) and it intercepts an arc of \((8x-10)^{\circ}\). Find \(x\), the angle, and the arc.
Problem 7: Triangle in a semicircle
Segment \(\overline{AB}\) is a diameter and \(C\) is on the circle. If \(\angle A=34^{\circ}\), find \(\angle ACB\) and \(\angle B\).
Problem 8: Semicircle algebra
Segment \(\overline{AB}\) is a diameter of a circle and \(C\) is another point on the circle. \(m\angle CAB=(2x+4)^{\circ}\) and \(m\angle CBA=(3x+6)^{\circ}\). Find \(x\) and both angles.
Problem 9: Angles on the same arc
Arc \(AB\) measures \(82^{\circ}\). Points \(C\) and \(D\) lie on the major arc. Find \(m\angle ACB\) and \(m\angle ADB\).
Problem 10: Radius and tangent
Line \(AP\) is tangent to the circle at \(A\). The radius \(OA\) is 6 and \(AP\) is 8. Find \(OP\).
Problem 11: Angle at a tangent
Line \(\ell\) is tangent to a circle with center \(O\) at point \(A\), and \(P\) is another point on \(\ell\). If \(m\angle AOP=52^{\circ}\), find \(m\angle APO\).
Problem 12: Two tangents
Tangent segments \(PA\) and \(PB\) touch a circle with center \(O\) at \(A\) and \(B\). Minor arc \(AB\) measures \(130^{\circ}\). Find \(m\angle AOB\) and \(m\angle APB\).
Problem 13: Equal tangent segments
From point \(P\) outside a circle with center \(O\), tangent segments \(PA\) and \(PB\) are drawn. The radius is 5 and \(OP=13\). Find \(PA\) and \(PB\).
Problem 14: Distance to a chord
A chord of length 24 is drawn in a circle of radius 13. Find the distance from the center to the chord.
Problem 15: Chord from a distance
A chord is 8 units from the center of a circle with radius 17. Find the length of the chord.
Problem 16: Equal chords
Chords \(AB\) and \(CD\) of the same circle have equal length. Arc \(AB\) measures \((4x+6)^{\circ}\) and arc \(CD\) measures \((6x-14)^{\circ}\). Find \(x\) and the arc measures.
Problem 17: Inscribed quadrilateral
Quadrilateral \(ABCD\) is inscribed in a circle. \(m\angle A=105^{\circ}\) and \(m\angle B=82^{\circ}\). Find \(m\angle C\) and \(m\angle D\).
Problem 18: Inscribed quadrilateral algebra
In an inscribed quadrilateral \(ABCD\), \(m\angle A=(4x+10)^{\circ}\) and \(m\angle C=(2x+20)^{\circ}\). Find \(x\) and both angles.
Problem 19: Ferris wheel seats
Aisha rides a Ferris wheel with 12 seats spaced equally around the rim. (a) Find the central angle between two neighboring seats. (b) Seats \(P\) and \(Q\) are 3 seats apart along the shorter way around. Find the central angle \(\angle POQ\). (c) Aisha sits in a seat on the longer arc between \(P\) and \(Q\). Find the angle between her lines of sight to \(P\) and \(Q\).
Problem 20: Pizza cut
Jamal cuts a circular pizza with a 20-inch diameter along a straight chord that is 12 inches long. How far is the cut from the center of the pizza?
Problem 21: Plaza lights
Light poles \(A\) and \(B\) stand at opposite ends of a diameter of a circular plaza that is 80 feet across. Liam stands at point \(L\) on the edge, 48 feet from pole \(A\). Find \(m\angle ALB\) and the distance from Liam to pole \(B\).
Problem 22: Pond path
A straight path touches a circular pond at point \(T\). The pond has radius 30 yards and its center is \(O\). Mei stands at \(P\) on the path, 40 yards from \(T\). Find \(OP\) and the shortest distance from Mei to the water.
Problem 23: Central and inscribed algebra
Central angle \(\angle AOB\) and inscribed angle \(\angle ACB\) intercept the same arc. \(m\angle AOB=(12x-20)^{\circ}\) and \(m\angle ACB=(5x-5)^{\circ}\). Find \(x\) and both angles.
Problem 24: Tangents and arcs algebra
Tangent segments from \(P\) touch a circle at \(A\) and \(B\). \(m\angle P=(4x-2)^{\circ}\) and minor arc \(AB\) measures \((6x+2)^{\circ}\). Find \(x\), the angle, and the arc.
See every chapter on the Geometry math page.