1.03f Circle properties: angles, chords, tangents

5 \includegraphics[max width=\textwidth, alt={}, center]{aa4c496d-ce5f-4f46-ad37-d901644a9e7c-06_761_460_258_840} The diagram shows a cord going around a pulley and a pin. The pulley is modelled as a circle with centre \(O\) and radius 5 cm . The thickness of the cord and the size of the pin \(P\) can be neglected. The pin is situated 13 cm vertically below \(O\). Points \(A\) and \(B\) are on the circumference of the circle such that \(A P\) and \(B P\) are tangents to the circle. The cord passes over the major arc \(A B\) of the circle and under the pin such that the cord is taut. Calculate the length of the cord.

8 \includegraphics[max width=\textwidth, alt={}, center]{80a20f05-61db-42d9-b4ba-53eea2290b2d-10_780_814_264_662} The diagram shows a symmetrical metal plate. The plate is made by removing two identical pieces from a circular disc with centre \(C\). The boundary of the plate consists of two \(\operatorname { arcs } P S\) and \(Q R\) of the original circle and two semicircles with \(P Q\) and \(R S\) as diameters. The radius of the circle with centre \(C\) is 4 cm , and \(P Q = R S = 4 \mathrm {~cm}\) also.

1. Show that angle \(P C S = \frac { 2 } { 3 } \pi\) radians.
2. Find the exact perimeter of the plate.
3. Show that the area of the plate is \(\left( \frac { 20 } { 3 } \pi + 8 \sqrt { 3 } \right) \mathrm { cm } ^ { 2 }\).

10 The equation of a circle is \(x ^ { 2 } + y ^ { 2 } - 4 x + 6 y - 77 = 0\).

1. Find the \(x\)-coordinates of the points \(A\) and \(B\) where the circle intersects the \(x\)-axis.
2. Find the point of intersection of the tangents to the circle at \(A\) and \(B\).

10 Points \(A ( - 2,3 ) , B ( 3,0 )\) and \(C ( 6,5 )\) lie on the circumference of a circle with centre \(D\).

1. Show that angle \(A B C = 90 ^ { \circ }\).
2. Hence state the coordinates of \(D\).
3. Find an equation of the circle.
   The point \(E\) lies on the circumference of the circle such that \(B E\) is a diameter.
4. Find an equation of the tangent to the circle at \(E\).

12 \includegraphics[max width=\textwidth, alt={}, center]{77f27b11-b931-481f-b4ef-5e549eff8086-18_1006_938_269_591} The diagram shows a circle \(P\) with centre \(( 0,2 )\) and radius 10 and the tangent to the circle at the point \(A\) with coordinates \(( 6,10 )\). It also shows a second circle \(Q\) with centre at the point where this tangent meets the \(y\)-axis and with radius \(\frac { 5 } { 2 } \sqrt { 5 }\).

1. Write down the equation of circle \(P\).
2. Find the equation of the tangent to the circle \(P\) at \(A\).
3. Find the equation of circle \(Q\) and hence verify that the \(y\)-coordinates of both of the points of intersection of the two circles are 11.
4. Find the coordinates of the points of intersection of the tangent and circle \(Q\), giving the answers in surd form.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

8 A circle with equation \(x ^ { 2 } + y ^ { 2 } - 6 x + 2 y - 15 = 0\) meets the \(y\)-axis at the points \(A\) and \(B\). The tangents to the circle at \(A\) and \(B\) meet at the point \(P\). Find the coordinates of \(P\). \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-10_71_1659_466_244} \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-10_2723_37_136_2010}

8 The points \(A ( 7,1 ) , B ( 7,9 )\) and \(C ( 1,9 )\) are on the circumference of a circle.

1. Find an equation of the circle.
2. Find an equation of the tangent to the circle at \(B\).

9 Functions f, g and h are defined as follows: $$\begin{aligned} & \mathrm { f } : x \mapsto x - 4 x ^ { \frac { 1 } { 2 } } + 1 \quad \text { for } x \geqslant 0 \\ & \mathrm {~g} : x \mapsto m x ^ { 2 } + n \quad \text { for } x \geqslant - 2 , \text { where } m \text { and } n \text { are constants, } \\ & \mathrm { h } : x \mapsto x ^ { \frac { 1 } { 2 } } - 2 \quad \text { for } x \geqslant 0 . \end{aligned}$$

1. Solve the equation \(\mathrm { f } ( x ) = 0\), giving your solutions in the form \(x = a + b \sqrt { c }\), where \(a , b\) and \(c\) are integers.
2. Given that \(\mathrm { f } ( x ) \equiv \mathrm { gh } ( x )\), find the values of \(m\) and \(n\). \includegraphics[max width=\textwidth, alt={}, center]{05e75fa2-81ae-44b1-b073-4100f5d911e0-16_652_1045_255_550} The diagram shows a circle with centre \(A\) of radius 5 cm and a circle with centre \(B\) of radius 8 cm . The circles touch at the point \(C\) so that \(A C B\) is a straight line. The tangent at the point \(D\) on the smaller circle intersects the larger circle at \(E\) and passes through \(B\).

9 A circle has centre at the point \(B ( 5,1 )\). The point \(A ( - 1 , - 2 )\) lies on the circle.

1. Find the equation of the circle.
   Point \(C\) is such that \(A C\) is a diameter of the circle. Point \(D\) has coordinates (5, 16).
2. Show that \(D C\) is a tangent to the circle.
   The other tangent from \(D\) to the circle touches the circle at \(E\).
3. Find the coordinates of \(E\).

11 A circle with centre \(C\) has equation \(( x - 8 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 100\).

1. Show that the point \(T ( - 6,6 )\) is outside the circle.
   Two tangents from \(T\) to the circle are drawn.
2. Show that the angle between one of the tangents and \(C T\) is exactly \(45 ^ { \circ }\).
   The two tangents touch the circle at \(A\) and \(B\).
3. Find the equation of the line \(A B\), giving your answer in the form \(y = m x + c\).
4. Find the \(x\)-coordinates of \(A\) and \(B\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

12 \includegraphics[max width=\textwidth, alt={}, center]{10b2ec29-adca-4313-ae24-bab8b2d9f8a4-18_750_981_258_580} The diagram shows the circle with equation \(x ^ { 2 } + y ^ { 2 } - 6 x + 4 y - 27 = 0\) and the tangent to the circle at the point \(P ( 5,4 )\).

1. The tangent to the circle at \(P\) meets the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\). Find the area of triangle \(O A B\), where \(O\) is the origin.
2. Points \(Q\) and \(R\) also lie on the circle, such that \(P Q R\) is an equilateral triangle. Find the exact area of triangle \(P Q R\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

8 \includegraphics[max width=\textwidth, alt={}, center]{8eb3d21b-dc45-493c-9e5c-3c0535c505e8-10_492_888_255_625} The diagram shows two identical circles intersecting at points \(A\) and \(B\) and with centres at \(P\) and \(Q\). The radius of each circle is \(r\) and the distance \(P Q\) is \(\frac { 5 } { 3 } r\).

1. Find the perimeter of the shaded region in terms of \(r\).
2. Find the area of the shaded region in terms of \(r\).

7 \includegraphics[max width=\textwidth, alt={}, center]{cbcb15b4-1870-4dfd-b6e9-839aa4601511-3_545_759_269_694} The diagram shows a circle with centre \(O\) and radius 8 cm . Points \(A\) and \(B\) lie on the circle. The tangents at \(A\) and \(B\) meet at the point \(T\), and \(A T = B T = 15 \mathrm {~cm}\).

1. Show that angle \(A O B\) is 2.16 radians, correct to 3 significant figures.
2. Find the perimeter of the shaded region.
3. Find the area of the shaded region.

5 \includegraphics[max width=\textwidth, alt={}, center]{b24ed4c7-ab07-45f4-adf2-027734c36b62-2_586_682_1726_733} In the diagram, \(O A B\) is a sector of a circle with centre \(O\) and radius 12 cm . The lines \(A X\) and \(B X\) are tangents to the circle at \(A\) and \(B\) respectively. Angle \(A O B = \frac { 1 } { 3 } \pi\) radians.

1. Find the exact length of \(A X\), giving your answer in terms of \(\sqrt { } 3\).
2. Find the area of the shaded region, giving your answer in terms of \(\pi\) and \(\sqrt { } 3\).

5 \includegraphics[max width=\textwidth, alt={}, center]{d71002bb-b6f0-42a3-89fb-f2769d5c3779-2_543_883_1274_630} The diagram shows a circle with centre \(O\) and radius 5 cm . The point \(P\) lies on the circle, \(P T\) is a tangent to the circle and \(P T = 12 \mathrm {~cm}\). The line \(O T\) cuts the circle at the point \(Q\).

1. Find the perimeter of the shaded region.
2. Find the area of the shaded region.

9 \includegraphics[max width=\textwidth, alt={}, center]{53839c8c-07ea-4545-9c00-a6884aa2afc3-3_387_1175_1781_486} In the diagram, \(O A B\) is an isosceles triangle with \(O A = O B\) and angle \(A O B = 2 \theta\) radians. Arc \(P S T\) has centre \(O\) and radius \(r\), and the line \(A S B\) is a tangent to the \(\operatorname { arc } P S T\) at \(S\).

1. Find the total area of the shaded regions in terms of \(r\) and \(\theta\).
2. In the case where \(\theta = \frac { 1 } { 3 } \pi\) and \(r = 6\), find the total perimeter of the shaded regions, leaving your answer in terms of \(\sqrt { } 3\) and \(\pi\).

7 \includegraphics[max width=\textwidth, alt={}, center]{d68c82ec-8c85-40b9-8e81-bd53c7f8dafe-3_462_956_258_593} In the diagram, \(A B\) is an arc of a circle, centre \(O\) and radius 6 cm , and angle \(A O B = \frac { 1 } { 3 } \pi\) radians. The line \(A X\) is a tangent to the circle at \(A\), and \(O B X\) is a straight line.

1. Show that the exact length of \(A X\) is \(6 \sqrt { } 3 \mathrm {~cm}\). Find, in terms of \(\pi\) and \(\sqrt { } 3\),
2. the area of the shaded region,
3. the perimeter of the shaded region.

6 \includegraphics[max width=\textwidth, alt={}, center]{58d65166-2b1a-4b58-9859-afe919c0a3a9-08_454_684_255_726} The diagram shows points \(A\) and \(B\) on a circle with centre \(O\) and radius \(r\). The tangents to the circle at \(A\) and \(B\) meet at \(T\). The shaded region is bounded by the minor \(\operatorname { arc } A B\) and the lines \(A T\) and \(B T\). Angle \(A O B\) is \(2 \theta\) radians.

1. In the case where the area of the sector \(A O B\) is the same as the area of the shaded region, show that \(\tan \theta = 2 \theta\).
2. In the case where \(r = 8 \mathrm {~cm}\) and the length of the minor \(\operatorname { arc } A B\) is 19.2 cm , find the area of the shaded region.

9

1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{0f58de6c-aba7-4a79-a962-c23be3ee0aa9-4_433_476_264_872} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} In Fig. 1, \(O A B\) is a sector of a circle with centre \(O\) and radius \(r\). \(A X\) is the tangent at \(A\) to the arc \(A B\) and angle \(B A X = \alpha\).
   1. Show that angle \(A O B = 2 \alpha\).
   2. Find the area of the shaded segment in terms of \(r\) and \(\alpha\).
2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{0f58de6c-aba7-4a79-a962-c23be3ee0aa9-4_451_503_1162_861} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} In Fig. 2, \(A B C\) is an equilateral triangle of side 4 cm . The lines \(A X , B X\) and \(C X\) are tangents to the equal circular \(\operatorname { arcs } A B , B C\) and \(C A\). Use the results in part (a) to find the area of the shaded region, giving your answer in terms of \(\pi\) and \(\sqrt { } 3\).
   [0pt] [6]

9 \includegraphics[max width=\textwidth, alt={}, center]{73c0c113-8f35-4e7f-ad5d-604602498b71-4_837_1020_255_559} The diagram shows two circles, \(C _ { 1 }\) and \(C _ { 2 }\), touching at the point \(T\). Circle \(C _ { 1 }\) has centre \(P\) and radius 8 cm ; circle \(C _ { 2 }\) has centre \(Q\) and radius 2 cm . Points \(R\) and \(S\) lie on \(C _ { 1 }\) and \(C _ { 2 }\) respectively, and \(R S\) is a tangent to both circles.

1. Show that \(R S = 8 \mathrm {~cm}\).
2. Find angle \(R P Q\) in radians correct to 4 significant figures.
3. Find the area of the shaded region.

6 \includegraphics[max width=\textwidth, alt={}, center]{3fd0b68f-41b1-4eee-8018-bcaf3cf22950-3_801_1273_255_434} The diagram shows a circle \(C _ { 1 }\) touching a circle \(C _ { 2 }\) at a point \(X\). Circle \(C _ { 1 }\) has centre \(A\) and radius 6 cm , and circle \(C _ { 2 }\) has centre \(B\) and radius 10 cm . Points \(D\) and \(E\) lie on \(C _ { 1 }\) and \(C _ { 2 }\) respectively and \(D E\) is parallel to \(A B\). Angle \(D A X = \frac { 1 } { 3 } \pi\) radians and angle \(E B X = \theta\) radians.

1. By considering the perpendicular distances of \(D\) and \(E\) from \(A B\), show that the exact value of \(\theta\) is \(\sin ^ { - 1 } \left( \frac { 3 \sqrt { } 3 } { 10 } \right)\).
2. Find the perimeter of the shaded region, correct to 4 significant figures.

6 \includegraphics[max width=\textwidth, alt={}, center]{3a631b88-5ba5-49e7-a312-dfd8a6d8a24e-2_615_809_1535_667} The diagram shows a metal plate \(A B C D\) made from two parts. The part \(B C D\) is a semicircle. The part \(D A B\) is a segment of a circle with centre \(O\) and radius 10 cm . Angle \(B O D\) is 1.2 radians.

1. Show that the radius of the semicircle is 5.646 cm , correct to 3 decimal places.
2. Find the perimeter of the metal plate.
3. Find the area of the metal plate.

4 \includegraphics[max width=\textwidth, alt={}, center]{518bb805-5b14-4b41-94fd-38a31a90c218-06_401_698_255_721} The diagram shows a semicircle with centre \(O\) and radius 6 cm . The radius \(O C\) is perpendicular to the diameter \(A B\). The point \(D\) lies on \(A B\), and \(D C\) is an arc of a circle with centre \(B\).

1. Calculate the length of the \(\operatorname { arc } D C\).
2. Find the value of \(\frac { \text { area of region } P } { \text { area of region } Q }\),
   giving your answer correct to 3 significant figures.

8 \includegraphics[max width=\textwidth, alt={}, center]{0e4a249a-9e6a-49d4-996c-fe07b7730f59-12_560_574_258_781} The diagram shows a sector \(O A C\) of a circle with centre \(O\). Tangents \(A B\) and \(C B\) to the circle meet at \(B\). The arc \(A C\) is of length 6 cm and angle \(A O C = \frac { 3 } { 8 } \pi\) radians.

1. Find the length of \(O A\) correct to 4 significant figures.
2. Find the perimeter of the shaded region.
3. Find the area of the shaded region.

4 \includegraphics[max width=\textwidth, alt={}, center]{567c3d72-c633-4ae0-8605-f63f93d718c4-06_517_768_262_685} The diagram shows a circle with centre \(O\) and radius \(r \mathrm {~cm}\). Points \(A\) and \(B\) lie on the circle and angle \(A O B = 2 \theta\) radians. The tangents to the circle at \(A\) and \(B\) meet at \(T\).

1. Express the perimeter of the shaded region in terms of \(r\) and \(\theta\).
2. In the case where \(r = 5\) and \(\theta = 1.2\), find the area of the shaded region.