Triangle and sector combined - area/perimeter with given values

CAIE P1 2021 June Q5

6 marks Standard +0.3

5 The diagram shows a triangle $A B C$, in which angle $A B C = 90 ^ { \circ }$ and $A B = 4 \mathrm {~cm}$. The sector $A B D$ is part of a circle with centre $A$. The area of the sector is $10 \mathrm {~cm} ^ { 2 }$.

Find angle $B A D$ in radians.
Find the perimeter of the shaded region.

CAIE P1 2022 June Q9

8 marks Standard +0.3

9 The diagram shows triangle $A B C$ with $A B = B C = 6 \mathrm {~cm}$ and angle $A B C = 1.8$ radians. The arc $C D$ is part of a circle with centre $A$ and $A B D$ is a straight line.

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

Edexcel C12 2016 October Q8

8 marks Standard +0.3

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{53865e15-3838-4551-b507-fe49549b87db-20_545_1048_212_584} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} The compound shape $A B C D A$, shown in Figure 1, consists of a triangle $A B D$ joined along its edge $B D$ to a sector $D B C$ of a circle with centre $B$ and radius 6 cm . The points $A , B$ and $C$ lie on a straight line with $A B = 5 \mathrm {~cm}$ and $B C = 6 \mathrm {~cm}$. Angle $D A B = 1.1$ radians.

Show that angle $A B D = 1.20$ radians to 3 significant figures.
Find the area of the compound shape, giving your answer to 3 significant figures.

Edexcel C2 2005 January Q7

11 marks Standard +0.3

7. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{13bca882-27da-40f2-99d8-4fdeb6629c4e-12_707_1072_301_434}

\end{figure} Figure 1 shows the triangle $A B C$, with $A B = 8 \mathrm {~cm} , A C = 11 \mathrm {~cm}$ and $\angle B A C = 0.7$ radians. The $\operatorname { arc } B D$, where $D$ lies on $A C$, is an arc of a circle with centre $A$ and radius 8 cm . The region $R$, shown shaded in Figure 1, is bounded by the straight lines $B C$ and $C D$ and the $\operatorname { arc } B D$. Find

the length of the $\operatorname { arc } B D$,
the perimeter of $R$, giving your answer to 3 significant figures,
the area of $R$, giving your answer to 3 significant figures.

Edexcel C2 2008 June Q7

12 marks Moderate -0.3

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0454f5f6-b5ee-40b1-bc6a-ff8aeb06a455-09_817_1029_205_484} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows $A B C$, a sector of a circle with centre $A$ and radius 7 cm .
Given that the size of $\angle B A C$ is exactly 0.8 radians, find

the length of the arc $B C$,
the area of the sector $A B C$. The point $D$ is the mid-point of $A C$. The region $R$, shown shaded in Figure 1, is bounded by $C D , D B$ and the arc $B C$. Find
the perimeter of $R$, giving your answer to 3 significant figures,
the area of $R$, giving your answer to 3 significant figures.

Edexcel C2 2010 June Q6

9 marks Standard +0.3

6. \includegraphics[max width=\textwidth, alt={}, center]{571780c2-945b-4636-b7c3-0bd558d28710-07_458_809_258_569} \section*{Figure 1} Figure 1 shows the sector $O A B$ of a circle with centre $O$, radius 9 cm and angle 0.7 radians.

Find the length of the arc $A B$.
Find the area of the sector $O A B$. The line $A C$ shown in Figure 1 is perpendicular to $O A$, and $O B C$ is a straight line.
Find the length of $A C$, giving your answer to 2 decimal places. The region $H$ is bounded by the arc $A B$ and the lines $A C$ and $C B$.
Find the area of $H$, giving your answer to 2 decimal places.
\section*{LU}

Edexcel C2 Specimen Q5

7 marks Moderate -0.3

5. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{afaf76d8-2a1f-4239-8275-70fad4f418c1-1_351_663_1529_693}

\end{figure} The shaded area in Fig. 1 shows a badge $A B C$, where $A B$ and $A C$ are straight lines, with $A B = A C = 8 \mathrm {~mm}$. The curve $B C$ is an arc of a circle, centre $O$, where $O B = O C =$ 8 mm and $O$ is in the same plane as $A B C$. The angle $B A C$ is 0.9 radians.

Find the perimeter of the badge.
Find the area of the badge.

OCR C2 2007 June Q8

11 marks Standard +0.3

8 \includegraphics[max width=\textwidth, alt={}, center]{e429080f-8634-46bc-b451-7b13b871e518-3_300_744_1046_703} The diagram shows a triangle $A B C$, where angle $B A C$ is 0.9 radians. $B A D$ is a sector of the circle with centre A and radius AB .

The area of the sector $B A D$ is $16.2 \mathrm {~cm} ^ { 2 }$. Show that the length of $A B$ is 6 cm .
The area of triangle $A B C$ is twice the area of sector $B A D$. Find the length of $A C$.
Find the perimeter of the region $B C D$.

OCR C2 2013 June Q5

8 marks Standard +0.3

5 \includegraphics[max width=\textwidth, alt={}, center]{b2c1188d-881e-4fb5-bece-5a51006543c7-2_405_688_1535_685} The diagram shows a sector $B A C$ of a circle with centre $A$ and radius 16 cm . The angle $B A C$ is 0.8 radians. The length $A D$ is 7 cm .

Find the area of the region $B D C$.
Find the perimeter of the region $B D C$.

AQA C2 2010 January Q1

9 marks Moderate -0.3

1 The diagram shows a sector $O A B$ of a circle with centre $O$. \includegraphics[max width=\textwidth, alt={}, center]{961ff4d6-b62a-4fab-8204-8a33a969d343-2_444_373_541_804} The radius of the circle is 15 cm and angle $A O B = 1.2$ radians.

1. Show that the area of the sector is $135 \mathrm {~cm} ^ { 2 }$.
2. Calculate the length of the arc $A B$.
The point $P$ lies on the radius $O B$ such that $O P = 10 \mathrm {~cm}$, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{961ff4d6-b62a-4fab-8204-8a33a969d343-2_449_378_1436_799} Calculate the perimeter of the shaded region bounded by $A P , P B$ and the arc $A B$, giving your answer to three significant figures.
(5 marks)

WJEC Unit 3 2022 June Q3

Moderate -0.3

The diagram below shows a plan of the patio Eric wants to build.

\includegraphics[max width=\textwidth, alt={}]{72bb1603-edbd-4e2e-bf2b-f33bb667e61b-2_517_746_1505_632}

The walls $O A$ and $O C$ are perpendicular. The straight line $A B$ is of length 4 m and is perpendicular to $O A$. The shape $O B C$ is a sector of a circle with centre $O$ and radius OC.
The angle $B O C$ is $\frac { \pi } { 3 }$ radians. Calculate the area of the patio $O A B C$. Give your answer correct to 2 decimal places. The sum to infinity of a geometric series with first term $a$ and common ratio $r$ is 120 . The sum to infinity of another geometric series with first term $a$ and common ratio $4 r ^ { 2 }$ is $112 \frac { 1 } { 2 }$. Find the possible values of $r$ and the corresponding values of $a$.

0

5

The function $f ( x )$ is defined by $$f ( x ) = \frac { 6 x + 4 } { ( x - 1 ) ( x + 1 ) ( 2 x + 3 ) }$$ a) Express $f ( x )$ in terms of partial fractions.
b) Find $\int \frac { 3 x + 2 } { ( x - 1 ) ( x + 1 ) ( 2 x + 3 ) } \mathrm { d } x$, giving your answer in the form $a \ln | g ( x ) |$, where $a$ is a real number and $g ( x )$ is a function of $x$.

0

6

Geraint opens a savings account. He deposits $\pounds 10$ in the first month. In each subsequent month, the amount he deposits is 20 pence greater than the amount he deposited in the previous month.
a) Find the amount that Geraint deposits into the savings account in the 12th month.
b) Determine the number of months it takes for the total amount in the savings account to reach $\pounds 954$.
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0
The diagram below shows a sketch of the curves $y = x ^ { 2 }$ and $y = 8 \sqrt { x }$. \includegraphics[max width=\textwidth, alt={}, center]{72bb1603-edbd-4e2e-bf2b-f33bb667e61b-3_508_869_2094_623} Find the area of the region bounded by the two curves.

CAIE P1 2011 November Q4

6 marks Moderate -0.3

\includegraphics{figure_4} In the diagram, $ABCD$ is a parallelogram with $AB = BD = DC = 10$ cm and angle $ABD = 0.8$ radians. $APD$ and $BQC$ are arcs of circles with centres $B$ and $D$ respectively.

Find the area of the parallelogram $ABCD$. [2]
Find the area of the complete figure $ABQCDP$. [2]
Find the perimeter of the complete figure $ABQCDP$. [2]

Edexcel C2 Q7

11 marks Standard +0.3

\includegraphics{figure_1} Figure 1 shows the triangle $ABC$, with $AB = 8$ cm, $AC = 11$ cm and $\angle BAC = 0.7$ radians. The arc $BD$, where $D$ lies on $AC$, is an arc of a circle with centre $A$ and radius 8 cm. The region $R$, shown shaded in Figure 1, is bounded by the straight lines $BC$ and $CD$ and the arc $BD$. Find

the length of the arc $BD$, [2]
the perimeter of $R$, giving your answer to 3 significant figures, [4]
the area of $R$, giving your answer to 3 significant figures. [5]

Edexcel C2 Q4

12 marks Moderate -0.3

\includegraphics{figure_2} Triangle $ABC$ has $AB = 9$ cm, $BC = 10$ cm and $CA = 5$ cm. A circle, centre $A$ and radius 3 cm, intersects $AB$ and $AC$ at $P$ and $Q$ respectively, as shown in Fig. 3.

Show that, to 3 decimal places, $\angle BAC = 1.504$ radians. [3]

Calculate,

the area, in cm$^2$, of the sector $APQ$, [2]
the area, in cm$^2$, of the shaded region $BPQC$, [3]
the perimeter, in cm, of the shaded region $BPQC$. [4]

Edexcel C2 Q6

12 marks Moderate -0.3

\includegraphics{figure_2} Triangle $ABC$ has $AB = 9$ cm, $BC = 10$ cm and $CA = 5$ cm. A circle, centre $A$ and radius 3 cm, intersects $AB$ and $AC$ at $P$ and $Q$ respectively, as shown in Fig. 2.

Show that, to 3 decimal places, $\angle BAC = 1.504$ radians. [3]

Calculate,

the area, in cm$^2$, of the sector $APQ$, [2]
the area, in cm$^2$, of the shaded region $BPQC$, [3]
the perimeter, in cm, of the shaded region $BPQC$. [4]

SPS SPS SM Pure 2023 June Q6

5 marks Moderate -0.3

\includegraphics{figure_2} The shape $AOCBA$, shown in Figure 2, consists of a sector $AOB$ of a circle centre $O$ joined to a triangle $BOC$. The points $A$, $O$ and $C$ lie on a straight line with $AO = 7.5$ cm and $OC = 8.5$ cm. The size of angle $AOB$ is 1.2 radians. Find, in cm, the perimeter of the shape $AOCBA$, giving your answer to one decimal place. [5]