GCE O-Level · Mathematics (4052)

Pythagoras’ theorem and trigonometry: Practice Questions

5 multiple-choice questions marked as you go, and 5 written questions with worked solutions. All on Pythagoras’ theorem and trigonometry.

10 questions26 marksFree, no account

Question 1

1 mark

In a right-angled triangle \(XYZ\), the lengths of the two shorter sides are \(XY = 5\) cm and \(YZ = 12\) cm, where \(\angle XYZ = 90^\circ\). Find the length of the hypotenuse \(XZ\).

Question 2

1 mark

In triangle \(PQR\), \(PQ = 7\) cm, \(QR = 9\) cm, and \(\angle PQR = 60^\circ\). Calculate the length of the side \(PR\) correct to two decimal places.

Question 3

1 mark

A ship sails from point \(A\) to point \(B\) on a bearing of \(065^\circ\). It then sails from \(B\) to \(C\) on a bearing of \(155^\circ\). Given that \(AB = 8\text{ km}\) and \(BC = 6\text{ km}\), calculate the bearing of \(A\) from \(C\) to the nearest degree.

Question 4

1 mark

In triangle \(ABC\), the length of side \(AB = 8\) cm, the length of side \(BC = 5\) cm, and \(\angle ABC = 30^\circ\). Calculate the area of triangle \(ABC\).

Question 5

1 mark

A vertical flagpole \( OT \) stands at the centre \( O \) of a horizontal rectangular field \( ABCD \). The length of the field \( AB \) is \( 16\text{ m} \) and the width \( BC \) is \( 12\text{ m} \). If the angle of elevation of the top of the flagpole \( T \) from the corner \( A \) is \( 35^\circ \), calculate the height of the flagpole \( OT \), giving your answer correct to 3 significant figures.

Question 6

2 marks

In the diagram, \(ABC\) is a right-angled triangle where \(\angle ABC = 90^\circ\), \(AB = 12\text{ cm}\), and \(BC = 5\text{ cm}\). Find the exact value of \(\sin(\angle BAC)\).

Write your answer out first, then check it against the worked solution.

Question 7

4 marks

In the diagram, \(D\), \(E\), and \(F\) are three points on horizontal ground. \(DE = 14\text{ m}\), \(EF = 9\text{ m}\), and \(\angle DEF = 62^\circ\).
(a) Calculate the distance \(DF\), giving your answer correct to \(3\) significant figures.
(b) Calculate the area of triangle \(DEF\), giving your answer correct to \(3\) significant figures.

Write your answer out first, then check it against the worked solution.

Question 8

6 marks

A hiker walks from point \(A\) on a bearing of \(040^\circ\) for \(5 \text{ km}\) to reach point \(B\). From \(B\), he then walks on a bearing of \(130^\circ\) to point \(C\). Given that point \(C\) is directly East of point \(A\), calculate the distance \(BC\).

Write your answer out first, then check it against the worked solution.

Question 9

4 marks

A vertical flagpole \(TF\) of height \(15\text{ m}\) stands on horizontal ground. A point \(G\) on the ground is \(20\text{ m}\) away from the base \(F\) of the flagpole.
(a) Calculate the distance from \(G\) to the top of the flagpole \(T\).
(b) Calculate the angle of elevation of the top of the flagpole \(T\) from the point \(G\), giving your answer correct to \(1\) decimal place.
(c) Write down the exact value of \(\tan(\angle TGF)\) as a simplified fraction.

Write your answer out first, then check it against the worked solution.

Question 10

5 marks

In the diagram, \(A\), \(B\), and \(C\) are three points on a horizontal plane. \(B\) is due east of \(A\). The distance \(AB = 14\text{ km}\), \(BC = 9\text{ km}\), and \(\angle ABC = 115^\circ\).
(a) Calculate the distance \(AC\), giving your answer correct to \(3\) significant figures.
(b) Calculate the area of triangle \(ABC\), giving your answer correct to \(1\) decimal place.
(c) Calculate \(\angle BAC\), giving your answer correct to \(1\) decimal place.
(d) Hence, find the bearing of \(C\) from \(A\).

Write your answer out first, then check it against the worked solution.

* The content provided by thinka is generated by AI and may not always be accurate or up-to-date. Please use it as a supplementary resource and verify with official materials.

You've seen the model answer. Now get yours marked.

This page can show you how a good answer looks. It cannot tell you what your answer was missing. thinka marks your written work against the real mark scheme in about 15 seconds.

Get My Answers Marked Back to the notes

Want more questions like these? Get a fresh set on this topic, marked as you go.