Circle theorems and constructionsIB MYP Maths Extended: Subtopic test

1. 1
   $A$, $B$ and $C$ are points on the circumference of a circle with centre $O$. $C$ is on the major arc $AB$ and angle $AOB=116^\circ$.

   (a)

   Find the size of angle $ACB$.

   [1 mark]

   • A$116^\circ$
   • B$232^\circ$
   • C$29^\circ$
   • D$58^\circ$

   (b)

   Find the size of angle $OAB$.

   [1 mark]

   • A$58^\circ$
   • B$32^\circ$
   • C$64^\circ$
   • D$29^\circ$

   (c)

   $D$ is a point on the minor arc $AB$. Find the size of angle $ADB$, giving a reason.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   $PQ$ is a diameter of a circle with centre $O$. $R$ is a point on the circle and angle $RPQ=35^\circ$. The line $QT$ is the tangent to the circle at $Q$, and $T$ is on the same side of $PQ$ as $R$.

   (a)

   Find the size of angle $PQR$.

   [1 mark]

   • A$35^\circ$
   • B$90^\circ$
   • C$55^\circ$
   • D$145^\circ$

   (b)

   Which reason shows that angle $PQT=90^\circ$?

   [1 mark]

   • AA tangent is perpendicular to the radius at the point of contact
   • BThe angle in a semicircle is $90^\circ$
   • CAngles in the same segment are equal
   • DOpposite angles of a cyclic quadrilateral add up to $180^\circ$

   (c)

   Find the size of angle $RQT$, giving a reason for each step.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Two tangents $TA$ and $TB$ are drawn from a point $T$ outside a circle with centre $O$. They touch the circle at $A$ and $B$. Angle $ATB=50^\circ$.

   (a)

   Find the size of angle $TAB$. Give a reason for each step.

   [3 marks]

   (b)

   Find the size of angle $AOB$ and the size of angle $OAB$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Three lamp-posts $A$, $B$ and $C$ stand on the edge of a circular pond. A planner wants to find the centre $O$ of the pond. $C$ is on the major arc $AB$ and angle $ACB=40^\circ$.

   (a)

   Describe how the planner can find the centre of the pond using only a compass and a ruler, and explain why the method works.

   [6 marks]

   (b)

   Once the planner has found $O$:
   (i) find angle $AOB$, giving a reason;
   (ii) find angle $OAB$, explaining your working;
   (iii) a fourth lamp-post $E$ is placed on the minor arc $AB$. Find angle $AEB$, giving a reason.

   [6 marks]

   Total for question 4: 12 marks

End of questions