T2T3 Review — Circle Geometry — Grade 10

Circle theorem angles. Fluency

(a) Central angle = 112°. Find the inscribed angle on the major arc.
(b) Inscribed angle = 47°. Find the central angle.
(c) AB is a diameter. Angle BAC = 28°. Find angle ABC.
(d) ABCD cyclic. Angle A = 73°, angle B = 88°. Find angles C and D.

Chords and tangents. Fluency

(a) Circle r=15 cm. Chord is 9 cm from the centre. Find the chord length.
(b) Two tangents from external point, one length = 11 cm. Find the other.
(c) Tangent ET=9 cm. Secant: near point EA=3 cm. Find EB.
(d) Chords intersect inside circle: AX=4, XC=9, BX=6. Find XD.

Arc length and sector area. Fluency

(a) r=9 cm, θ=80°. Find arc length.
(b) r=7 m, θ=270°. Find sector area.
(c) Sector area = 25π cm², r=10 cm. Find θ.
(d) r=6 cm, θ=120°. Find the minor segment area to 2 d.p.

Mixed calculations. Fluency

(a) External point P, OP=17 cm, r=8 cm. Find tangent length PT.
(b) r=13 cm, chord at distance 5 cm from centre. Find chord length.
(c) Arc length = 8π cm, θ = 160°. Find r.
(d) Alternate segment: tangent-chord angle = 52°. Find the inscribed angle in the alternate segment.

Angles in a circle from a diagram. Understanding

The diagram shows a circle with centre O. Points A, B, C lie on the circle. AB is a chord (not a diameter). Angle AOB = 100°.

(a) Find angle ACB (C on the major arc). State the theorem.
(b) D is another point on the major arc. Find angle ADB.
(c) E is on the minor arc. Find angle AEB.
(d) ACBD is a cyclic quadrilateral. Find angle ADB + angle ACB.

Tangent and chord combined. Understanding

From external point P, a tangent touches the circle at T and a secant passes through A (near) and B (far). PA = 5 cm, AB = 20 cm, OT ⊥ PT (as expected).

(a) Find PT.
(b) Find the radius of the circle if OP = 15 cm.
(c) Find angle TPO if OP = 15 and r = 10.
(d) A chord AB passes through the midpoint of OP. The midpoint of OP is M. Using the power of point M, find AM×MB if OT = 10 and OP = 15 so OM = 7.5. (Power of point M = OM² − r² if M is outside... or r² − OM² if inside.)

Sector perimeter and area. Understanding

(a) A sector has perimeter 30 cm and radius 9 cm. Find the central angle.
(b) A sector has area 40 cm² and radius 8 cm. Find the central angle.
(c) A sector has arc length 10 cm and area 40 cm². Find r and θ.
(d) A sector and a triangle share the same base (chord AB) and height. The sector has r=10 and θ=60°. Find the ratio of sector area to triangle area.

Cyclic quadrilateral angles. Understanding

(a) ABCD cyclic. Angle A = (x+20)°, C = (2x−5)°. Find x.
(b) PQRS cyclic. Angle P : angle R = 2 : 3. Find both angles.
(c) A cyclic quadrilateral has one pair of opposite angles both equal to 90°. What shape must it be?
(d) A tangent at D makes an angle of 65° with chord DA. ABCD is cyclic. Find angle ABC.

Radar sweep. Problem Solving

A radar antenna at airport A sweeps a sector of radius 80 km with a central angle of 140° once every 4 seconds.

(a) Find the arc length of the radar sweep.
(b) Find the area covered by one sweep.
(c) The beam sweeps back and forth (140° left, then 140° right). Find the total area scanned in 1 minute.
(d) An aircraft must remain within the radar sector at all times. It is currently 60 km from A at the extreme left of the sector (at the arc edge, bearing at the left boundary). It flies directly toward A at 900 km/h. How long before it exits the sector from the opposite side?

Garden fountain design. Problem Solving

A circular garden pond (radius 3 m) has a tangent path running alongside it. The path touches the pond at point T. A straight bridge starts at external point P on the path (where PT = 4 m) and crosses the pond as a chord.

(a) Find the distance from P to the centre O of the pond.
(b) The bridge (secant from P) enters the pond at A (PA = 2 m) and exits at B. Find PB.
(c) Find the chord length AB.
(d) Find the distance from the centre O to the chord (the bridge).

Tangent lines and circle distance. Problem Solving

Two circles, each of radius 5 cm, have their centres 13 cm apart. A common external tangent touches both circles.

(a) Explain why the tangent is perpendicular to both radii at the tangent points.
(b) The two radii to the tangent points are parallel (equal circles, external tangent). Draw the perpendicular from O⊂2; to the radius O⊂1;T⊂1;. Find the length of this perpendicular.
(c) Find the length of the common external tangent segment (T⊂1;T⊂2;).
(d) Find the length of the common internal tangent (crosses between the circles).

Sector optimisation. Problem Solving

A sector of fixed perimeter P = 20 cm is to enclose the maximum possible area.

(a) Write the perimeter in terms of r and θ (in degrees).
(b) Express θ in terms of r.
(c) Write the area as a function of r alone.
(d) By testing r = 3, 4, 5, 6, 7, find the radius that maximises area.

Topic Review — Circle Geometry