Right-Angled Triangles - Pythagoras & Trigonometry

The diagram shows a quadrilateral .

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

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

 = 1.7 m
 = 1.5 : 1

The angle of elevation of  from  is 52°.

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

 is a trapezium.

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

 is a right-angled triangle.
is a point on .

Angle = 30°
= 10.4 cm
= 5.2 cm
= 18 cm

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

 is a trapezium.

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

 = 15 m
 = 24 m
Angle  = 62°

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

The diagram shows a triangle  inside a rectangle .

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

Here is a parallelogram.

 = 7cm
 = 5 cm
Angle  is 40°

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

Show that the value of       is an integer.

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 in the diagram.

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

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

Work out the least possible height of the post
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 that is metres from the base of the cliff.
The angle of elevation from to the top of the cliff is 20°

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

......................................................

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

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

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

The diagram shows a shaded shape made from a semicircle and a right-angled triangle .

is the diameter of the semicircle .

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

....................

The diagram shows four congruent right‐angled triangles  and .
The diagram also shows the straight line .

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

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

Write down the exact value of tan 60°.

Find the exact area of this triangle.

..................... cm²

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.

Trianglehas perimeter 20 cm.

= 7 cm.
= 4 cm.

By calculation, deduce whether triangle 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 is about 75°.

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

is a right-angled triangle.

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

The length of the side is reduced by 1 cm.

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

Will the value of cosincrease or decrease?
You must give a reason for your answer.

The diagram shows the positions of three turbines  and .

 is 6 km due north of turbine .
is 4.5 km due west of turbine .

Calculate the distance .

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

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° sin60°
Give your answer in its simplest form.

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

The diagram shows isosceles triangle 

             

Calculate the area of triangle

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

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

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

The diagram shows a trapezium.

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

and are points on a circle, centre .
is a diameter of the circle.




Angle

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

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

From point , Stanley walks  due east to point .
From , he then walks  due south to point .

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

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

 is a trapezium.

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

The diagram shows an isosceles triangle.

Work out the area of the triangle.

.........................

 is a rectangle.
 is a straight line.

 = 12 cm
Angle = 60°
Angle  = 90°

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

and  are points on the circumference of a circle, centre .
 is a diameter of the circle.

Explain why angle  is 

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

The diagram represents a metal frame.

The frame is made from four metal bars,  and .

Angle = angle  = 90°
 = 5 m
 = 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  and  that satisfy the equation 

Find the value of  when = 45.

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,  cm, of the ladder.

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

Simplify   

Give your answer in the form   where and 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.

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

Angle

Show that .

is a point on the wheel.

Angle

Work out the height of above horizontal ground.

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

Here is a sketch of the graph for one complete turn of the wheel.

 is the highest point on the graph.

Work out the coordinates of .

Here is triangle 

The point  lies on .

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

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

In triangle

The points and lie on so that is the diameter of the semicircular arc
The radius of the semicircular arc is 3cm.

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

ADB and BCD are right-angled triangles.
BC = CD.

Angle BAD = 30°.

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

Without using a calculator, show that .

The point A is shown on the unit grid below.
The point B is units from A and lies on the intersection of two grid lines.

Mark one possible position for B.