Pythagoras & Trigonometry (Edexcel GCSE Maths: Higher): Exam Questions

$ABC$is a right-angled triangle. $AC$= 6 cm $AB$  = 13 cm

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

Here is a rectangle.

The 8-sided shape below is made from 4 of these rectangles and 4 congruent right-angled triangles.

Work out the perimeter of the 8-sided shape. You must show all your working.

Here is part of a field.

This part of the field is in the shape of a trapezium. A farmer wants to put a fence all the way around the edge of this part of the field.

The farmer has 50m of fence.

Does he have enough fence? You must show all your working.

The diagram shows a rectangular framework.

The framework is made from 5 metal rods. The metal rods have a weight of 0.9 kg per metre.

Work out the total weight of the framework. Give your answer, in kg, correct to 3 significant figures.

Triangle$ABC$has perimeter 20 cm.

$AB$= 7 cm. $BC$= 4 cm.

By calculation, deduce whether triangle $ABC$is a right-angled triangle.

The diagram shows a ladder leaning against a vertical wall.

The ladder stands on horizontal ground.

The length of the ladder is 6 m. The bottom of the ladder is 2.25 m from the bottom of the wall.

A ladder is safe to use when the angle marked $y$ is about 75°.

Is the ladder safe to use? You must show all your working.

$ABC$is a right-angled triangle.

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

The length of the side $AB$is reduced by 1 cm.

The length of the side $BC$is still 7 cm. Angle $ACB$is still 90°

Will the value of cos$ABC$increase or decrease? You must give a reason for your answer.

The diagram shows the positions of three turbines $A,B$ and $C$.

$A$ is 6 km due north of turbine $B$. $C$ is 4.5 km due west of turbine $B$.

Calculate the distance $AC$.

Calculate the bearing of $C$ from $A$. Give your answer correct to the nearest degree.

$ABC$is an isosceles triangle.

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

Find the exact value of tan30° $\times$ sin60°

Give your answer in its simplest form.

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

The diagram shows isosceles triangle $ABC$

$AB=AC=17.5\mathrm{cm}$             $BC=28\mathrm{cm}$

Calculate the area of triangle $ABC$

...................................................... cm²

The diagram shows two vertical phone masts, $AB$ and $CD$, on horizontal ground.

$AB=6.2\mathrm{m}$      $AC=244\mathrm{m}$     $CD=30.7\mathrm{m}$

Work out the size of the angle of depression of $B$ from $D$ Give your answer correct to one decimal place.

The diagram shows a trapezium.

Work out the value of $y.$ Give your answer correct to $1$ decimal place.

$y=$

$M,N$ and $P$ are points on a circle, centre $O$.
$MON$ is a diameter of the circle.

$MP=3.5cm$
$PN=9.7cm$

Angle $MPN=90°$

Work out the circumference of the circle.
Give your answer correct to 3 significant figures.

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

From point $A$, Stanley walks $200\mathrm{m}$ due east to point $B$. From $B$, he then walks $160\mathrm{m}$ due south to point $C$.

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

.......................................metres

$ABCD$ is a trapezium.

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

The diagram shows an isosceles triangle.

Work out the area of the triangle.

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

Calculate the exact value of $2\sin \left(45\right)+3\cos \left(45\right)$.

Ted wants to build a single storey extension on the back of his house. The diagram shows a sketch of his plan.

The outer wall is 4 metres from the house wall.

Both walls are parallel to each other and perpendicular to the floor.

The height of the outer wall is 2 metres.

The perpendicular distance from the floor of the house wall to the windowsill above is 3.85 metres.

The slope of the roof must have a minimum angle of elevation of $20°$ to allow for drainage.

The roof joins the outer wall to the house wall and must meet the house wall below the windowsill.

Show that the extension roof will fit on the back of the house with the current measurements.

The diagram below shows the position of two vertical flag poles, $AB$ and $CD$, on horizontal ground.

$CD$ is 2.2 m.

$AB:CD=1:1.2$.

The angle of elevation of $D$ from $B$ is $35°$

Calculate the length of $AC$.

Give your answer correct to 3 significant figures.

Triangles $ABD$ and $BCD$ are right-angled triangles.

Work out the value of $x$.
Give your answer correct to 2 decimal places.

The diagram shows a quadrilateral $ABCD$.

$$\begin{gathered} AB=16\mathrm{cm}. \\ AD=12\mathrm{cm}. \\ \mathrm{Angle}BCD=40°. \\ \mathrm{Angle}ADB=\mathrm{angle}CBD=90°. \end{gathered}$$

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

The diagram shows two vertical posts, $AB$ and $CD$, on horizontal ground.

$AB$ = 1.7 m
$CD:AB$ = 1.5 : 1

The angle of elevation of $C$ from $A$ is 52°.

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

$ABCD$ is a trapezium.

$$\begin{gathered} AD=10\mathrm{cm} \\ AB=9\mathrm{cm} \\ DC=3\mathrm{cm} \\ \mathrm{Angle}ABC=\mathrm{angle}BCD=90° \end{gathered}$$

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

$ABC$ is a right-angled triangle.
$D$ is a point on $AB$.

Angle $ACD$ = 30°
$AD$= 10.4 cm
$DB$= 5.2 cm
$AC$= 18 cm

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

$ABCD$ is a trapezium.

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

$AB$ = 15 m
$BC$ = 24 m
Angle $BAD$ = 62°

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

The diagram shows a triangle $DEF$ inside a rectangle $ABCD$.

Show that the area of triangle $DEF$ is 8 cm²
You must show all your working.

Here is a parallelogram.

$DC$ = 7cm
$CB$ = 5 cm
Angle $ABC$ is 40°

Work out the area of the parallelogram.
Give your answer correct to 1 decimal place.

The diagram shows a right-angled triangle.

Five of these triangles are put together to make a shape.

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

A zip wire is shown as the dashed line $AC$ in the diagram.

The zip wire is supported by two vertical posts $AB$ and $CD$ standing on horizontal ground.

$CD=2.6\mathrm{m}$         $BD=12\mathrm{m}$

The zip wire makes an angle $x$ with the horizontal, as shown in the diagram.
The design of the zip wire requires the angle $x$ to be at least 5°

Work out the least possible height of the post $AB$
Give your answer correct to 3 significant figures.

The diagram shows a vertical cliff with a vertical radio mast on top of the cliff and a buoy in the sea.

The height of the cliff is 100 metres.
The buoy is at the point $B$ that is $d$metres from the base of the cliff.
The angle of elevation from $B$ to the top of the cliff is 20°

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

$d=$ ......................................................

The point $A$ at the top of the radio mast is vertically above the top of the cliff.
The angle of elevation from $B$ to $A$ is 25°

Calculate the height of the radio mast.
Give your answer correct to 3 significant figures.

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

The diagram shows a shaded shape $ABCD$made from a semicircle $ABC$and a right-angled triangle $ACD$.

$AC$is the diameter of the semicircle $ABC$.

Work out the perimeter of the shaded shape.
Give your answer correct to 3 significant figures.

....................$\mathrm{cm}$

The diagram shows four congruent right‐angled triangles $ABJ,BCI,CDH$ and $DEG$.
The diagram also shows the straight line $ABCDEF$.

$AJ=15\mathrm{cm}$
$\mathrm{Angle}BAJ=35°$

$AF=80\mathrm{cm}$

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

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

Show that the value of    $5\sin 30°\times \cos 30°\times 8\tan 30°$   is an integer.

Write down the exact value of tan 60°.

Find the exact area of this triangle.

..................... cm²

The following circle has centre $F$.

Line $AB$ is a tangent to the circle at point $C$.

Line $BDF$ is a straight line.

The length of $BC=15$ cm.

Calculate the exact length of $DF$.

Triangle $ABC$ is inscribed in a square $ADEC$. The square $ADEC$ is inscribed in a circle with centre $O$.

The radius of the circle $OC$ is 4cm.

Calculate the area of triangle $ABC$.

$PQRS$ is a trapezium.

A square has the same area as this trapezium.

Calculate the perimeter of the square.

Give your answer to 3 significant figures.

ABCD is a parallelogram made up of two congruent right-angled triangles.

All measurements are in centimetres.

The area of parallelogram ABCD is 30 ${\mathrm{cm}}^2$.

Find the perimeter of the parallelogram ABCD.

Give your answer correct to 3 significant figures.

A square, with sides of length $x$ cm, is inside a circle.
Each vertex of the square is on the circumference of the circle.

The area of the circle is 49 cm².

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

$ABCD$ is a rectangle.
$CDE$ is a straight line.

$AB$ = 12 cm
Angle $ACB$= 60°
Angle $EAC$ = 90°

Calculate the length of $CE$.
You must show all your working.

$B,C$ and $D$ are points on the circumference of a circle, centre $O$. $BOD$ is a diameter of the circle.

$AO=7\mathrm{cm}\mathrm{Angle}ABO=90°\mathrm{Angle}OAB=35°\mathrm{Angle}DBC=70°$

Explain why angle $BCD$ is $90°$

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

The diagram represents a metal frame.

The frame is made from four metal bars, $AB,AC,BC$ and $BD$.

Angle $ABC$ = angle $ADB$ = 90°
$AB$ = 5 m
$BC$ = 3 m

Work out the total length of the four metal bars of the frame. Give your answer correct to 3 significant figures.

The table shows some values of $x$ and $y$ that satisfy the equation $y=a\cos x°+b$

Find the value of $y$ when $x=45$

Here is triangle $ABD.$

The point $C$ lies on $BD$.

$AD=13\mathrm{cm}BC=8\mathrm{cm}\mathrm{angle}ADB=90°\mathrm{angle}CAD=20°$

Calculate the size of angle $BAC$.
Give your answer correct to $1$ decimal place.

In the diagram, $ABC$ is a right-angled triangle and $DEF$ is a semicircular arc.

In triangle $ABC$

The points $D$ and $F$ lie on $AC$ so that $DF$ is the diameter of the semicircular arc $DEF$
The radius of the semicircular arc is 3cm.

Work out the length of $AFEDC$
Give your answer correct to 2 significant figures.

The diagram shows the side view of a step ladder with a horizontal strut of length 48 cm.

The strut is one third of the way up the ladder.

The symmetrical cross section of the ladder shows two similar triangles.

Work out the vertical height, $h$ cm, of the ladder.

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

Simplify   $\frac{2\sin 45°-\tan 45°}{4\tan 60°}$

Give your answer in the form  $\frac{\sqrt{a}-\sqrt{b}}{c}$ where $a,b$ and $c$ are integers.

A Big Wheel is modelled as a circle with centre O and radius 15 metres.

The wheel turns in an anticlockwise direction.

The lowest point on the wheel is always 2 metres above horizontal ground.

$C$ is a point on the wheel, $h$ metres above horizontal ground.

Angle $COB=x°$
Show that $h=17-15\cos x°$

$D$ is a point on the wheel.

Angle $DOB=120°$

Work out the height of $D$ above horizontal ground.

........................ metres

Here is a sketch of the graph $h=17-15\cos x°$for one complete turn of the wheel.

$P$ is the highest point on the graph.

Work out the coordinates of $P$.

ADB and BCD are right-angled triangles.
BC = CD.
$\mathrm{AD}=10\sqrt{6}\mathrm{mm}$

Angle BAD = 30°.

$\tan 30°=\frac{1}{\sqrt{3}}$

Work out the length of BC.

...........................mm

A, B, C and D are points on the circumference of a circle, centre O.

Angle CAD = 28° and CD = 6.4cm.
BD is a diameter of the circle.

Calculate the area of the circle.

...................... cm²

Show that $\sqrt{20}=2\sqrt{5}$.

The point A is shown on the unit grid below.
The point B is $2\sqrt{5}$ units from A and lies on the intersection of two grid lines.

Mark one possible position for B.