Circle geometry

Learn

1. 1The angle at the centreWhy the angle an arc makes at the centre is always twice the angle it makes on the circle, and how to use it.Done12 min
2. 2Same segment and semicircleAngles in the same segment are equal, and the angle in a semicircle is 90°. Both come straight from lesson 1; here you learn to spot them in a busy diagram.Done12 min
3. 3Cyclic quadrilateralsWhen all four corners of a quadrilateral are on a circle, opposite angles add up to 180° and each exterior angle equals the interior opposite angle.Done12 min
4. 4TangentsA tangent meets the radius at 90°, two tangents from a point are equal, and the angle between a tangent and a chord equals the angle in the alternate segment.Done15 min
5. 5Solving exam questionsA routine for any circle question, then two WAEC theory questions that need several theorems in a row.Done15 min

Past questions (146)

From 1983–2025: 76 WAEC, 13 NECO, 57 JAMB. Theory questions first, then objective, newest first.

Theory (53)

• NECO 2024 · Paper 2 · Q11Construct a quadrilateral $ABCD$ such that $|AB| = 6.0\text{ cm}$, $|DB| = 7.5\text{ cm}$, $\angle DAB = 60^\circ$, $|DC| = 7.0\text{ cm}$ and …
• WAEC 2024 · Paper 2 · Q12In the diagram, $PR$ is a tangent to the circle centre $O$ at $Q$, $\angle POQ = 56^\circ$, and $PO$ meets the chord $SQ$ at $V$ such that $\angle SVP = 109^\circ$. …
• WAEC 2024 · Paper 2 · Q13In the diagram, $O$ is the centre of the circle, $PQ$ is a tangent to the circle at $T$ and $ABC$ is a straight line. $TC$ bisects $\angle BTQ$, …
• WAEC 2024 · Paper 2 · Q4In the diagram, $O$ is the centre of the circle and $OBCD$ is a rhombus. $\angle ADO = \angle OBA = y$ and $\angle BAD = t$. Find:
• WAEC 2024 · Paper 2 · Q9The diameter of a circle centre $O$ is $26\text{ cm}$. A chord $PQ$ is drawn such that its midpoint is $5\text{ cm}$ from $O$. Calculate, correct …
• NECO 2023 · Paper 2 · Q3In the diagram, $O$ is the centre of the circle, the chord $AB$ is $12\text{ cm}$ long and $\angle ACB = 30^\circ$.
• WAEC 2023 · Paper 2 · Q12In the diagram, $P$, $Q$, $R$ and $S$ are points on the circle with centre $O$; $QR \parallel OS$, $\angle QOR = 2m$, $\angle QPR = n$ and $\angle SOR = 54^\circ$. …
• WAEC 2023 · Paper 2 · Q4In the diagram, $P$, $Q$, $R$ and $S$ are points on the circle centre $K$. $\overline{KR}$ is a bisector of $\angle SRQ$, $\angle KSP = 41^\circ$ …
• WAEC 2023 · Paper 2 · Q3An isosceles triangle $PQR$ has its vertices on the circumference of a circle. If $|PQ| = |QR| = 17\text{ cm}$, $|PR| = 16\text{ cm}$ and $M$ …
• NECO 2022 · Paper 2 · Q5In the figure below, $O$ is the centre of the circle, $|AB| = |CD|$ and $\angle COD = 85^\circ$. Find the following:
• WAEC 2022 · Paper 2 · Q11The exterior angles of a polygon are $42^\circ, 38^\circ, 57^\circ, x^\circ, (x + y)^\circ, (2x - 15)^\circ$ and $(3x - y)^\circ$. If $x$ is …
• WAEC 2022 · Paper 2 · Q4In the diagram, $BCDE$ is a circle with centre $A$. $\angle BCD = (2x + 40)^\circ$, $\angle BAD = (5x - 35)^\circ$, $\angle BED = (2y + 10)^\circ$ …
• WAEC 2022 · Paper 2 · Q12In the diagram, $W$, $X$, $Y$, $Z$ are points on a circle centre $O$. $\angle WXY = 111^\circ$, $\angle OYX = 43^\circ$ and $|WZ| = |YZ|$. Calculate:
• WAEC 2022 · Paper 2 · Q4$TU$ is a tangent to the circle $PQRS$ at $P$. $\angle SPU = 30^\circ$, $\angle QRS = 70^\circ$ and $|QR| = |RS|$. Find:
• WAEC 2021 · Paper 2 · Q12In the diagram, $PQRS$ is a circle. $|PQ| = |QS|$, $\angle SPR = 26^\circ$ and the interior angles of $\triangle PQS$ are in the ratio $2 : 3 : 3$. …
• WAEC 2021 · Paper 2 · Q13On Sam's first birthday celebration, his grandfather deposited an amount of $1,000.00 in a bank compounded at $4\%$ interest annually. Find how much is in the account if Sam is 4 years old.
• WAEC 2021 · Paper 2 · Q4In the diagram, $AD$ is a diameter of the circle with centre $O$, $C$ is a point on $AD$ produced, and $BC$ is a tangent to the circle at $B$. …
• WAEC 2021 · Paper 2 · Q11In the diagram, $O$ is the centre of the circle $PRSTU$ and $\angle PTS = 58^\circ$. Find: (i) $\angle POS$; (ii) $\angle PRS$.
• WAEC 2021 · Paper 2 · Q4$\overline{RU}$ and $\overline{RS}$ are tangents to the circle with centre $O$ at $U$ and $S$, and $T$ is a point on the major arc $US$. If $\angle URS = 80^\circ$, …
• WAEC 2020 · Paper 2 · Q11In the diagram, $MNPQ$ is a circle with centre $O$, and $MOQ$ is a diameter. $|MN| = |NP|$ and $\angle OMN = 50^\circ$. Find: (i) $\angle MNP$; …
• WAEC 2020 · Paper 2 · Q12In the diagram, $\overline{AB}$ is a tangent to the circle with centre $O$ and $COEB$ is a straight line. If $\overline{CD} \parallel \overline{AB}$ …
• WAEC 2020 · Paper 2 · Q3In the diagram, $O$ is the centre of the circle $ABCDE$, $BE$ and $AD$ are diameters, $|BC| = |CD|$ and $\angle BCD = 108^\circ$. Find $\angle CDE$.
• WAEC 2020 · Paper 2 · Q13The area of a triangle is $54\text{ cm}^2$. If the height is $3\text{ cm}$ more than the base, find the length of the base.
• WAEC 2020 · Paper 2 · Q4In the diagram, $R$, $S$, $T$, $U$ are points on a circle and $QRS$ is a straight line. $\angle STU = 124^\circ$ and $\angle QUR = 31^\circ$. …
• WAEC 2020 · Paper 2 · Q11In the diagram, $O$ is the centre of circle $ABCD$ such that $\angle AOB = 98^\circ$, $\angle DBA = 68^\circ$ and $\angle BDC = 47^\circ$. Find: …
• WAEC 2020 · Paper 2 · Q3An aeroplane flies $500\text{ km}$ from town $P$ on a bearing of $053^\circ$ to town $Q$. It then flies $700\text{ km}$ to town $R$ on a bearing …
• WAEC 2019 · Paper 2 · Q13In the diagram, $\overline{RS}$ and $\overline{RT}$ are tangents to the circle with centre $O$. $\angle TUS = 68^\circ$, $\angle SRT = x$ and …
• WAEC 2019 · Paper 2 · Q4If $\log_{10} a = 1.3010$ and $\log_{10} b = 1.4771$, find the value of $ab$.
• WAEC 2019 · Paper 2 · Q13In the diagram, $A$, $B$, $C$, $D$ are points on the circle with centre $O$. $\angle AOD = 130^\circ$ and $\angle BAO = 26^\circ$. Find: (i) …
• WAEC 2019 · Paper 2 · Q5In the diagram, $MNPR$ is a circle with centre $O$. The reflex angle at $O$ is $196^\circ$ and $\angle NMO = 52^\circ$. Find the value of $m$ …
• WAEC 2018 · Paper 2 · Q4In the diagram, $QOS$ is a diameter, $\angle RQS = x^\circ$ and $\angle QST = (3x + 15)^\circ$, where $RST$ is a straight line. Find: (i) the …
• WAEC 2018 · Paper 2 · Q4A man was charged 2 kobo per month for every ₦1.00 he borrowed from a bank. At what rate per annum was the interest charged?
• WAEC 2018 · Paper 2 · Q5Find the range of values of $x$ which satisfy the following inequalities simultaneously: $5 - x > 1$ and $9 + x \ge 8$.
• WAEC 2017 · Paper 2 · Q9$PQ$ is a tangent to a circle $RST$ at the point $S$. $PRT$ is a straight line, $\angle TPS = 34^\circ$ and $\angle TSQ = 65^\circ$. (i) Illustrate …
• WAEC 2017 · Paper 2 · Q4The diagram shows a circle centre $O$ and $\overline{PQ}$ is a diameter. $S$ and $R$ are points on the circle with $|SR| = |RQ|$. If $\angle PSO = 44^\circ$, …
• WAEC 2017 · Paper 2 · Q8The probabilities that Akafi and Iniola will pass an examination are $\frac34$ and $\frac35$ respectively. Find the probability that only Iniola …
• WAEC 2016 · Paper 2 · Q3In the diagram, $\angle RTS = 28^\circ$, $\angle VRM = 46^\circ$ and $MQ$ is a tangent to the circle $VRSTU$ at the point $R$. Find $\angle VUS$.
• WAEC 2016 · Paper 2 · Q7Using the completing the square method, solve, correct to 2 decimal places, $\dfrac{x - 2}{4} = \dfrac{x + 2}{2x}$.
• WAEC 2016 · Paper 2 · Q3If $x = -1$, $y = -3$, $z = -4$ and $w = -7$, evaluate $\dfrac{x^3 - y^2}{2w - z}$.
• WAEC 2015 · Paper 2 · Q11Make $m$ the subject of the relation $h = \dfrac{mt}{d(m + p)}$.
• WAEC 2015 · Paper 2 · Q3The ratio of the interior angle to the exterior angle of a regular polygon is $5 : 2$. Find the number of sides of the polygon.
• WAEC 2014 · Paper 2 · Q9In the Venn diagram, $P$, $Q$ and $R$ are subsets of the universal set $U$. The regions are: $P$ only $16 - 2x$, $P \cap Q$ only $5x$, $Q$ only …
• WAEC 2014 · Paper 2 · Q12In the diagram, $\overline{TS}$ is a tangent to the circle $PQRS$ at $S$, and $O$ is the centre of the circle ($QOS$ is a diameter). If $\angle TSP = 21^\circ$ …
• WAEC 2013 · Paper 2 · Q12An open rectangular tank is made from a steel plate of area $1440\text{ m}^2$. Its length is twice its width. If the depth of the tank is $4\text{ m}$ …
• WAEC 2013 · Paper 2 · Q5The pie chart shows the result of a survey conducted on the maintenance of five small and medium scale industries in a state. If the cost of …
• WAEC 2012 · Paper 2 · Q3In the diagram, $\overline{TU}$ is a tangent to the circle. $\angle RVU = 100^\circ$ and $\angle URS = 36^\circ$. Calculate the value of angle …
• WAEC 2011 · Paper 2 · Q4In the diagram, $PQRST$ is a quadrilateral. $PT \parallel QS$, $\angle PTQ = 42^\circ$, $\angle TSQ = 38^\circ$ and $\angle QSR = 30^\circ$. …
• WAEC 2011 · Paper 2 · Q8Given that $\sin x = 0.6$ and $0^\circ \le x \le 90^\circ$, evaluate $2\cos x + 3\sin x$, leaving your answer in the form $\frac mn$, where $m$ …
• WAEC 2010 · Paper 2 · Q3In the diagram, $PQ \parallel UT$, $\angle PQR = 125^\circ$, $\angle QRS = r$, $\angle RST = 80^\circ$ and $\angle STU = 44^\circ$. Calculate …
• WAEC 2010 · Paper 2 · Q4Find the value of $x$ if $x3_{\text{five}} - 14_{\text{five}} = 2x_{\text{five}}$.
• WAEC 2009 · Paper 2 · Q7A woman looking out from the window of a building at a height of 30 m observed that the angle of depression of the top of a flag pole was $44^\circ$. …
• WAEC 2009 · Paper 2 · Q5In the diagram, the two circles intersect at $X$ and $Y$. The centre $O$ of the smaller circle is on the circumference of the bigger circle. …
• WAEC 2008 · Paper 2 · Q11In the diagram, $AB \parallel CD$ and $BC \parallel FE$, $\angle CDE = 75^\circ$ and $\angle DEF = 26^\circ$. Find the angles marked $x$ and …

Objective (93)

• WAEC 2025 · Paper 1 · Q29A $120^\circ$ sector of a circle of radius $21\text{ cm}$ is bent to form a cone. What is the base radius of the cone?
• WAEC 2025 · Paper 1 · Q32A sector has angle $108^\circ$ and radius $3\frac12\text{ cm}$. Find its perimeter. $\left[\pi = \frac{22}{7}\right]$
• WAEC 2025 · Paper 1 · Q34In the diagram, $O$ is the centre of the circle, $AOD$ is produced to $C$ and $|BD| = |DC|$. If $\angle DCB = 35^\circ$, find $\angle BAO$.
• NECO 2024 · Paper 1 · Q34In the diagram, $PQS$ is a circle with centre $O$, $RST$ is a tangent at $S$ and $\angle SOP = 120^\circ$. Find $\angle PST$.
• NECO 2024 · Paper 1 · Q35In the diagram, $A$, $B$, $C$ and $D$ are points on a circle with centre $O$, and $BA$ is produced to $M$. If $\angle MAD = 82^\circ$ and $\angle ADO = 74^\circ$, …
• NECO 2024 · Paper 1 · Q40Calculate the length of an arc which subtends an angle of $66^\circ$ at the centre of a circle of radius $8\text{ cm}$, correct to one decimal …
• WAEC 2024 · Paper 1 · Q23In the diagram, $O$ is the centre of the circle. If $|\overline{OA}| = 25\text{ cm}$ and $|\overline{AB}| = 40\text{ cm}$, find $|\overline{OH}|$.
• WAEC 2024 · Paper 1 · Q30In the diagram, $\angle SQR = 52^\circ$ and $\angle PRT = 16^\circ$. Find the value of the angle marked $y$.
• WAEC 2024 · Paper 1 · Q31In the diagram, $JKL$ is a tangent to the circle $GHIK$ at $K$. $\angle LKG = 38^\circ$ and $\angle HIK = 87^\circ$. Calculate the value of the …
• WAEC 2024 · Paper 1 · Q45In the diagram, $TU$ is a tangent to the circle at $P$. If $\angle PTS = 44^\circ$ and $\angle SQP = 35^\circ$, find $\angle PST$.
• NECO 2023 · Paper 1 · Q35In the diagram, $O$ is the centre of the circle $PQR$. If $\angle QPR = 64^\circ$ and $\angle QRP = 46^\circ$, calculate $\angle POQ$.
• NECO 2023 · Paper 1 · Q40In the figure, $O$ is the centre of the circle and $DB$ is a diameter. $AED$ and $ABC$ are straight lines, $\angle EAB = 34^\circ$ and $\angle EDB = 40^\circ$. …
• NECO 2023 · Paper 1 · Q41In the diagram, $O$ is the centre of the circle, $AB$ is a diameter, $|DB| = |BC|$ and $\angle ABC = 54^\circ$. Find $\angle ACD$.
• NECO 2023 · Paper 1 · Q43In the diagram, $KTN$ is a tangent to the circle at $T$ and $\angle ABT = 65^\circ$. Find $\angle NTA$.
• WAEC 2023 · Paper 1 · Q17In the diagram, $O$ is the centre of the circle; $|NT| = |ST|$ and $\angle NTS = 36^\circ$. $RN$ and $RS$ are tangents at $N$ and $S$. Find the …
• WAEC 2023 · Paper 1 · Q30In the diagram, $NR$ is a diameter, $SRM$ is a straight line, $\angle MNR = x^\circ$ and $\angle SRN = (5x + 20)^\circ$. Find the value of $2x^\circ$.
• WAEC 2023 · Paper 1 · Q37In the diagram, $O$ is the centre of the circle and $\angle SQR = 28^\circ$. Find $\angle ORS$.
• WAEC 2023 · Paper 1 · Q46In the diagram, $M$, $N$, $R$ are points on the circle centre $O$; $\angle ORN = 48^\circ$ and $\angle RNM = 124^\circ$. Find $\angle OMN$.
• NECO 2022 · Paper 1 · Q34Find the value of $k$ in the figure, where $O$ is the centre of the circle.
• NECO 2022 · Paper 1 · Q44In the figure, $O$ is the centre of the circle. Find $\angle CAF$.
• NECO 2022 · Paper 1 · Q46In the figure, $PQ$ and $PR$ are tangents from $P$ to a circle with centre $O$. If $\angle QPR = 34^\circ$, find the angle marked $x$.
• WAEC 2022 · Paper 1 · Q30In the diagram, $MNR$ is a tangent to the circle at $N$ and $\angle NOS = 108^\circ$. Find $\angle OSN$.
• WAEC 2022 · Paper 1 · Q31In the diagram, $MNR$ is a tangent to the circle centre $O$ at $N$ and $\angle NOS = 108^\circ$. Find $\angle SNR$.
• WAEC 2022 · Paper 1 · Q35The diagram shows triangle $PQR$ inscribed in a circle. $PS$ is a tangent to the circle at $P$, $\angle QPR = 58^\circ$ and $\angle RPS = 73^\circ$. …
• WAEC 2022 · Paper 1 · Q36In the diagram, triangle $MNR$ is inscribed in circle $MNR$ and $PQ$ is a straight line. If $\angle MRN = 41^\circ$ and $\angle PMR = 141^\circ$, …
• WAEC 2021 · Paper 1 · Q29In the diagram, $O$ is the centre of the circle, $\angle PQR = 72^\circ$ and $OR$ is parallel to $PS$. Find $\angle OPS$.
• WAEC 2021 · Paper 1 · Q34In the diagram, $MP$ is a tangent to the circle at $N$, $\angle PNQ = 64^\circ$ and $|RQ| = |RN|$. Find the angle $t$.
• WAEC 2021 · Paper 1 · Q36In the cyclic quadrilateral $PQRS$, $\angle P = \angle S = x^\circ$, $\angle Q = (2y - 30)^\circ$ and $\angle R = (x + y)^\circ$. Find the value …
• WAEC 2021 · Paper 1 · Q39In the diagram, $EC$ is a diameter of the circle. If $\angle ABC = 158^\circ$, find $\angle ADE$.
• WAEC 2020 · Paper 1 · Q33In the diagram, $O$ is the centre of the circle and $RP$ is a diameter. If $\angle OPQ = 48^\circ$, find the value of $m$.
• WAEC 2020 · Paper 1 · Q36In the diagram, $O$ is the centre of the circle. $SOQ$ is the diameter and $\angle SRP = 37^\circ$. Find $\angle PSQ$.
• WAEC 2020 · Paper 1 · Q43In the diagram, $O$ is the centre of the circle. $PQ$ and $RS$ are tangents to the circle at the ends of a diameter. Find the value of $(m + n)^\circ$.
• WAEC 2020 · Paper 1 · Q44In the diagram, $O$ is the centre of the circle. If $\angle NLM = 74^\circ$, $\angle LMN = 39^\circ$ and $\angle LOM = x$, find the value of …
• WAEC 2019 · Paper 1 · Q38In the diagram, $O$ is the centre of the circle of radius $18\text{ cm}$. If $\angle ZXY = 70^\circ$, calculate the length of arc $ZY$. $\left[\text{Take }\pi = \frac{22}{7}\right]$
• WAEC 2019 · Paper 1 · Q39In the diagram, $RT$ is a tangent to the circle at $R$, $\angle PQR = 70^\circ$, $\angle QRT = 52^\circ$, $\angle QSR = y$ and $\angle PRQ = x$. …
• WAEC 2019 · Paper 1 · Q40In the same diagram ($RT$ a tangent at $R$, $\angle PQR = 70^\circ$, $\angle QRT = 52^\circ$), find the value of $x = \angle PRQ$.
• JAMB 2017 · UTME · Q12$PQRSTV$ is a regular hexagon of side $7\text{ cm}$ inscribed in a circle. Find the circumference of the circle. $\left[\text{Take }\pi = \frac{22}{7}\right]$
• JAMB 2017 · UTME · Q36In the figure, determine the angle marked $y$.
• JAMB 2017 · UTME (set 2) · Q13In the diagram, $R$, $S$ and $T$ lie on a circle with centre $O$. If $\angle RST = 60^\circ$, find the angle marked $x$.
• JAMB 2017 · UTME (set 2) · Q48In the diagram, $MN$ is a chord of a circle $KMN$ with centre $O$ and radius $10\text{ cm}$. If $\angle MON = 140^\circ$, find, correct to the …
• JAMB 2013 · UTME · Q28Find the length of a chord which subtends an angle of $90^\circ$ at the centre of a circle whose radius is $8\text{ cm}$.
• JAMB 2012 · UTME · Q27In the diagram, $PQR$ is a circle with centre $O$ and $PR$ a diameter. If $\angle QPR = x^\circ$, find $\angle QRP$.
• JAMB 2011 · UTME · Q29A chord of a circle of radius $7\text{ cm}$ is $5\text{ cm}$ from the centre of the circle. What is the length of the chord?
• JAMB 2010 · UTME · Q28In the diagram, $O$ is the centre of the circle, $ROS$ is a straight line produced to $U$, $UT$ is a tangent to the circle at $T$ and $\angle TRS = 25^\circ$. …
• JAMB 2010 · UTME · Q30From the cyclic quadrilateral $TUVW$, where $\angle TUV = (3x + 20)^\circ$ and $\angle TWV = 88^\circ$, find the value of $x$.
• JAMB 2004 · UME · Q25$PQRSTV$ is a regular polygon of side 7 cm inscribed in a circle. Find the circumference of the circle $PQRSTV$. $\left[\pi = \frac{22}{7}\right]$
• JAMB 2004 · UME · Q26$P$, $R$ and $S$ lie on a circle centre $O$ as shown, while $Q$ lies outside the circle. Find $\angle PSO$.
• JAMB 2003 · UME · Q23In the diagram, $O$ is the centre of the circle, $POM$ is a diameter and $\angle MNQ = 42^\circ$. Calculate $\angle QMP$.
• JAMB 2003 · UME · Q27In the diagram, $PQR$ is a straight line and $PS$ is a tangent to the circle $QRS$, with $PS = SR$ and $\angle SPR = 40^\circ$. Find $\angle PSQ$.
• JAMB 2003 · UME · Q34$XYZ$ is a circle with centre $O$ and radius 7 cm, and $\angle XZY = 45^\circ$. Find the area of the shaded segment cut off by $XY$. $\left[\pi = \frac{22}{7}\right]$
• JAMB 2002 · UME · Q10In the diagram, $XZ$ is the diameter of the circle $XYZ$ with centre $O$ and radius $\frac{15}{2}$ cm. If $XY = 12$ cm, find the area of the …
• JAMB 2001 · UME · Q25The chord $ST$ of a circle is equal to the radius $r$ of the circle. Find the length of the arc $ST$.
• JAMB 2000 · UME · Q27An equilateral triangle of side $\sqrt3$ cm is inscribed in a circle. Find the radius of the circle.
• JAMB 2000 · UME · Q29In the diagram, $\angle RPS = 50^\circ$, $\angle RPQ = 30^\circ$ and $PQ = QR$. Find the value of $\angle PRS$.
• JAMB 2000 · UME · Q30In the diagram, $EFGH$ is a circle with centre $O$. $FH$ is a diameter and $GE$ is a chord which meets $FH$ at right angles at $N$. If $NH = 8$ …
• JAMB 1999 · UME · Q28In the figure, $TZ$ is a tangent to the circle $QPZ$ and $TPQ$ is a straight line. Find $x = TP$ if $TZ = 6$ units and $PQ = 9$ units.
• JAMB 1999 · UME · Q31In the figure, $PQRS$ is a circle with $ST \parallel RQ$ ($T$ on $PQ$) and $PT = PS$. If $\angle QRS = 110^\circ$, find the value of $x = \angle PQR$.
• JAMB 1999 · UME · Q32In the diagram, $EFGH$ is a cyclic quadrilateral in which $EH \parallel FG$, and $FH$ and $EG$ are chords. If $\angle FHG = 42^\circ$ and $\angle EFH = 34^\circ$, …
• JAMB 1998 · UME · Q25In the diagram, $PR$ is a diameter of the circle $PQRS$, and $PST$ and $QRT$ are straight lines. If $\angle SPR = 35^\circ$ and $\angle PTQ = 30^\circ$, …
• JAMB 1998 · UME · Q28$TQ$ is a tangent to the circle $XYTR$. If $\angle YXT = 32^\circ$ and $\angle RTQ = 40^\circ$, find $\angle YTR$.
• JAMB 1998 · UME · Q29A chord of a circle of radius $\sqrt3$ cm subtends an angle of $60^\circ$ at the circumference. Find the length of the chord.
• JAMB 1997 · UME · Q25Each of the base angles of an isosceles triangle is $58^\circ$ and all the vertices lie on a circle. Determine the angle which the base of the …
• JAMB 1997 · UME · Q27The figure shows circles of radii $3\text{ cm}$ and $2\text{ cm}$ with centres $X$ and $Y$ respectively. The circles have a direct common tangent …
• JAMB 1995 · UME · Q26Determine the value of $x$ in the figure.
• JAMB 1995 · UME · Q27$PT$ is a tangent to the circle $TYZX$, $YT = YX$ and $\angle PTX = 50^\circ$. Calculate $\angle TZY$.
• JAMB 1994 · UME · Q27In the diagram, $O$ is the centre of the circle and $SOQ$ is a diameter. If $\angle PRS = 38^\circ$, what is the value of $\angle PSQ$?
• JAMB 1994 · UME · Q31In the diagram, $PTS$ is a tangent to the circle $TQR$ at $T$, $\angle QRT = 60^\circ$ and $\angle QTR = 50^\circ$. Calculate $\angle RTS$.
• JAMB 1993 · UME · Q28In the diagram, $PQRS$ is a circle with centre $O$ and diameter $SQ$, and $PQ \parallel RT$. If $\angle RTS = 32^\circ$, find $\angle PSQ$.
• JAMB 1993 · UME · Q29In the diagram, $O$ is the centre of the circle and $POQ$ is a diameter. If $\angle POR = 96^\circ$, find the value of $\angle ORQ$.
• JAMB 1993 · UME · Q31In the figure, $PT$ is a tangent to the circle at $U$ and $QU \parallel RS$. If $\angle TUR = 35^\circ$ and $\angle SRU = 50^\circ$, find $x = \angle QRU$.
• JAMB 1993 · UME · Q34In the figure, the line segment $ST$ is a tangent to the two circles at $S$ and $T$. $O$ and $Q$ are the centres of the circles with $OS = 5$ …
• JAMB 1992 · UME · Q26In the figure, $PQR$ is a semicircle, and $PQ$ and $QR$ are chords. $QS$ is the perpendicular from $Q$ to the diameter $PR$. What is the expression …
• JAMB 1991 · UME · Q35In the figure, $PMN$ and $PQR$ are two secants of the circle $MQTRN$ and $PT$ is a tangent. If $PM = 5$ cm, $PN = 12$ cm and $PQ = 4.8$ cm, calculate …
• JAMB 1991 · UME · Q36In the figure, $PMN$ and $PQR$ are two secants of the circle $MQTRN$ and $PT$ is a tangent. If $\angle PNR = 110^\circ$ and $\angle PMQ = 55^\circ$, …
• JAMB 1991 · UME · Q38In the figure, $PQ = PR = PS$, $QR$ is produced to $T$ and $\angle SRT = 68^\circ$. Find $\angle QPS$.
• JAMB 1990 · UME · Q33In the figure, $PS = QS = RS$ and $\angle QSR = 100^\circ$. Find $\angle QPR$.
• JAMB 1990 · UME · Q37In the figure, $PQRS$ is a circle and $PQT$ and $SRT$ are straight lines. If $\angle SPQ = 81^\circ$ and $\angle PTS = 22^\circ$, find $x = \angle PQR$.
• JAMB 1989 · UME · Q31$MN$ is a tangent to the circle at $M$, and $MR$ and $MQ$ are two chords, with $R$, $Q$ and $N$ on a straight line. If $\angle QMN = 60^\circ$ …
• JAMB 1988 · UME · Q30In the figure, $P$, $Q$, $R$ and $S$ lie on a circle and the chords $QR$ and $RS$ are equal. $SR$ is produced to $T$ with $\angle QRT = 120^\circ$, …
• JAMB 1988 · UME · Q37In the figure, $PS = RS = QS$ and $\angle QSR = 50^\circ$. Find $\angle QPR$.
• JAMB 1988 · UME · Q38In the figure, $XR$ and $YQ$ are tangents to the circle $YZXP$. If $\angle ZXR = 45^\circ$ and $\angle YZX = 55^\circ$, find $\angle ZYQ$.
• JAMB 1987 · UME · Q35Two chords $QR$ and $NP$ of a circle intersect inside the circle at $X$. If $\angle RQP = 37^\circ$, $\angle RQN = 49^\circ$ and $\angle QPN = 35^\circ$, …
• JAMB 1987 · UME · Q45What is the locus of the mid-points of all chords of length 6 cm within a circle of radius 5 cm and centre $O$?
• JAMB 1986 · UME · Q33In the diagram, $PQ$ and $RS$ are chords of a circle with centre $O$ which, produced, meet at $T$ outside the circle. If $TP = 24$ cm, $TQ = 8$ …
• JAMB 1986 · UME · Q37$PQ$ and $PR$ are tangents from $P$ to a circle with centre $O$. If $\angle QRP = 34^\circ$, find the angle marked $x$ ($\angle QOR$).
• JAMB 1985 · UME · Q37In the figure, $MNQP$ is a cyclic quadrilateral; $MN$ and $PQ$ are produced to meet at $X$, and $NQ$ and $MP$ are produced to meet at $Y$. If …
• JAMB 1985 · UME · Q40In the figure, $POQ$ is the diameter of the circle $PQRS$. If $\angle PSR = 145^\circ$, find $x^\circ$ (the angle $QPR$).
• JAMB 1985 · UME · Q43In the figure, $PQ$ is the tangent from $P$ to the circle $QRS$, with $SR$ a diameter and $SRP$ a straight line. If $\angle PQR = \theta$ and …
• JAMB 1984 · UME · Q46In the figure, $O$ is the centre of circle $PQRS$, $PR$ and $SQ$ are diameters, and $PS \parallel RT$. If $\angle PRT = 135^\circ$, find $\angle PSQ$.
• JAMB 1983 · UME · Q26If $O$ is the centre of the circle in the figure, find the value of $x$.
• JAMB 1983 · UME · Q36$PQRS$ is a cyclic quadrilateral in which $PQ = PS$. $PT$ is a tangent to the circle and $PQ$ makes an angle of $50^\circ$ with the tangent, …
• JAMB 1983 · UME · Q41In the figure, $O$ is the centre of the circle and the reflex angle $POQ$ is $235^\circ$. Find $\angle PRQ$.
• JAMB 1983 · UME · Q49In the figure, $PT$ is a tangent to the circle with centre $O$. If $\angle PQT = 30^\circ$, find the value of $\angle PTO$.