Sine, Cosine Rule & Area of Triangles (AQA GCSE Maths: Higher): Exam Questions

Work out the length of  $AD$.
Give your answer correct to 3 significant figures.

$ABC$is a triangle.

$$\begin{gathered} AB=8.7\mathrm{cm}. \\ \mathrm{Angle}ABC=49°. \\ \mathrm{Angle}ACB=64°. \end{gathered}$$

Calculate the area of triangle $ABC$.
Give your answer correct to $3$ significant figures.

The diagram shows triangle $LMN$.

Calculate the length of $LN$.
Give your answer correct to 3 significant figures.

$ABC$is a triangle.
$D$ is a point on $AC$.
Angle $BAD$= 45°
Angle $ADB$= 80°
$AB$ = 7.4cm
$DC$ = 5.8 cm

Work out the length of $BC$.
Give your answer correct to 3 significant figures.

$ABC$is a triangle.

$AC$= 8.4m
Angle $ACB$= 40°

The area of the triangle = 100 m².

Work out the length of $AB$.
Give your answer correct to 3 significant figures.
You must show all your working.

In triangle $RPQ$,

$RP$ = 8.7 cm
$PQ$ = 5.2 cm
Angle $PRQ$= 32°

Assuming that angle $PQR$is an acute angle,
calculate the area of triangle $RPQ$.
Give your answer correct to 3 significant figures.

If you did not know that angle $PQR$is an acute angle, what effect would this have on your calculation of the area of triangle $RPQ$?

The diagram shows the triangle $PQR$.

$$\begin{gathered} PQ=x\mathrm{cm} \\ PR=2x\mathrm{cm} \end{gathered}$$
Angle $QPR=30°$ 

The area of triangle $PQR=A{\mathrm{cm}}^2$

Show that   $x=\sqrt{2}A$

$ABC$is an isosceles triangle.

Work out the area of the triangle.
Give your answer correct to 3 significant figures.

A shape is made by joining triangle $ABC$to a semicircle with diameter $AC$.

Work out the total area of the shape.

Here is a triangular sail.

Vicky needs to buy waterproofing liquid for the sail.

She will put 3 coats of liquid on each side of the sail.
A litre of liquid covers 8.5 square metres of sail.

How many 1-litre bottles of liquid does Vicky need?

Another sail is joined to the first sail as shown.

$x$ is an acute angle.

Work out the size of angle $x$.

Here is a triangle $XYZ.$

The length $XZ$ and the angles $YXZ$ and $XYZ$are each given correct to 2 significant figures.

Calculate the upper bound for the length $YZ$.
Give your answer correct to one decimal place.
Show your working clearly.

....................................................... cm

Here is a quadrilateral $PQRS$.

Angle $SRQ$ is acute.

Work out the size of angle $SQR$.
Give your answer correct to $1$ decimal place.

The diagram shows triangle ABC with D on AC and E on AB.
DE is a straight line.

AD = 28 m, AE = 41 m, DE = 22 m and BC = 64 m.

Calculate the length CD.

....................................................... m

The diagram shows some land in the shape of a quadrilateral, ABCD.

AB = 3km, AD = 5km, CD = 12km and angle BAC = 30°.

The land is sold for £10 million per square kilometre.

Calculate the total cost of the land.

£ .............................................. million

Find $x$. You must show your working.

A cruise ship sets off from point A and travels on a bearing of $145°$ for 80 miles to reach point B.

Three days later, the cruise ship sets sail from point B and travels on a bearing of $060°$ to reach point C.

From point C, the cruise ship is 158 miles away from its starting location.

Find the distance between point B and C to the nearest mile.

The area of triangle $ABC$ is $6\sqrt{2}$m².

Calculate the value of $x$.
Give your answer correct to $3$significant figures.

$ABC$and $ADC$are triangles.

The area of triangle $ADC$is 56 m²

Work out the length of $AB.$
Give your answer correct to 1 decimal place.

$ABCD$is a parallelogram.

$AC$ = 9 cm
$DC$ = 11 cm
Angle $DAC$ = 100°

Calculate the area of the parallelogram.
Give your answer correct to 3 significant figures.

$ABC$is an acute-angled triangle.
$BA$= 7cm
$BC$  = 8cm

The area of triangle $ABC$ is 18 cm².

Work out the size of angle $BAC$.
Give your answer correct to 3 significant figures.
You must show all your working.

The diagram shows a hexagon $ABCDEF$.

$ABEF$and $CBED$are congruent parallelograms where $AB=BC=x\mathrm{cm}.$
$P$ is the point on $AF$and $Q$ is the point on $CD$such that $BP=BQ=10\mathrm{cm}.$

Given that angle $ABC=30°,$

prove that  $\cos PBQ=1-\frac{\left(2-\sqrt{3}\right)}{200}{x}^2$

$ABC$is an acute angled triangle.

The area of triangle $ABC$is 19 cm².

Work out the size of angle $ACB$.
Give your answer correct to 3 significant figures.

$ABC$is a triangle.
$D$ is a point on $AB$.

Work out the area of triangle $BCD$.
Give your answer correct to 3 significant figures.

In this question, all lengths are in centimetres.

$A$ is a point on a circle, centre $O$.
$B$ is a point on a different circle, centre $O$.
$AB=20$

The equation of the larger circle is    $x^2+y^2=144$
radius of smaller circle : radius of larger circle   = 4 : 5

Work out the size of angle $AOB$

............................................ degrees

The area of this triangle is $25\sqrt{3}{\mathrm{cm}}^2$

Work out the value of $w$.
Give your answer in the form $a\sqrt{b}$ where $a$ and $b$ are integers greater than 1.

The diagram shows a logo.

$ABE$ and $DCE$ are congruent triangles.
$BCE$ is a sector of a circle, centre $E$.

Show that the area of the logo is 510 cm² to 2 significant figures.

The diagram shows triangle$PQR$

Given that angle $PQR$is obtuse, work out the area of triangle $PQR$.
Give your answer correct to 3 significant figures.

The diagram shows triangle $ABC$

$c=11.5$correct to one decimal place
$x=80$     correct to the nearest whole number
$y=75$   correct to the nearest whole number

Calculate the upper bound for the value of $b$
Show your working clearly.
Give your answer correct to 3 significant figures.

$ABCD$ is a quadrilateral where $A,B,C$ and $D$ are points on a circle.

$AB=8\mathrm{cm}$
$BC=7.5\mathrm{cm}$
Angle $ABC=98°$
Angle $ACD=35°$

Work out the perimeter of quadrilateral $ABCD$.
Give your answer correct to one decimal place.

....................................................... cm

Here is a triangle $XYZ$.

The perimeter of the triangle is $k\mathrm{cm}$.

Given that $x=y-1$
find the value of $k$.
Show your working clearly.

$k=$                  

The diagram shows a regular octagon $ABCDEFGH.$

Each side of the octagon has length $10\mathrm{cm}.$

Find the area of the shaded region $ACDEH.$
Give your answer correct to the nearest ${\mathrm{cm}}^2$

............................................. ${\mathrm{cm}}^2$