Pythagoras and trigonometry in right-angled triangles – Year 11 GCSE maths worksheet with answers to download as PDF

Find the hypotenuse of each right-angled triangle. Give non-exact answers to 2 decimal places.

1. Shorter sides 9 cm and 12 cm. [2 marks]
2. Shorter sides 7 cm and 24 cm. [2 marks]
3. Shorter sides 5 cm and 8 cm. [2 marks]
4. What do you notice about the answers to (a) and (b)? [1 mark]

Find the missing side in each right-angled triangle.

1. Hypotenuse 17 cm, one side 8 cm. [2 marks]
2. Hypotenuse 10 cm, one side 7 cm. Give your answer to 2 decimal places. [2 marks]
3. Hypotenuse \(c\), sides 1.5 m and 2 m. Find \(c\). [2 marks]

A triangle is right-angled only if \(a^2+b^2=c^2\) for its longest side \(c\).

1. Is the triangle with sides 20 cm, 21 cm and 29 cm right-angled? [3 marks]
2. Is the triangle with sides 8 cm, 10 cm and 13 cm right-angled? [3 marks]
3. A builder checks a corner using 3 m, 4 m and 5 m. Explain why this works. [1 mark]

Give each answer to 2 decimal places.

1. A right-angled triangle has a hypotenuse of 12 cm and an angle of \(35^\circ\). Find the side opposite the angle. [3 marks]
2. Find the side adjacent to the \(35^\circ\) angle. [2 marks]
3. The side adjacent to a \(28^\circ\) angle is 9 cm. Find the hypotenuse. [3 marks]

Find the marked angle \(x\) in each right-angled triangle, to 1 decimal place.

1. Opposite 7 cm, adjacent 10 cm. [3 marks]
2. Opposite 5 cm, hypotenuse 9 cm. [3 marks]
3. Adjacent 4 cm, hypotenuse 11 cm. [3 marks]

Without a calculator, use the exact values you know.

1. Write the exact value of \(\tan60^\circ\). [2 marks]
2. A right-angled triangle has a \(30^\circ\) angle and hypotenuse 10 cm. Find the exact opposite and adjacent sides. [3 marks]
3. Find the exact area of the triangle. [2 marks]

A 5 m ladder makes an angle of \(70^\circ\) with the horizontal ground.

1. How high up the wall does it reach? [3 marks]
2. How far is the foot from the wall? [2 marks]
3. For safety the angle must be between \(65^\circ\) and \(75^\circ\). Is it safe? [1 mark]
4. The ladder is moved so that its foot is 1 m from the wall. What angle does it now make with the ground? [3 marks]

Mia stands 15 m from a tree. Her eyes are 1.6 m above the ground. The angle of elevation of the top of the tree is \(40^\circ\).

1. Sketch the situation and find the height of the tree above her eye level. [3 marks]
2. Find the total height of the tree. [1 mark]
3. How far is Mia's eye from the top of the tree? [2 marks]
4. Mia walks 10 m closer. What is the new angle of elevation? [3 marks]

A plane at a height of 3 000 m starts its descent towards the runway. The angle of depression of the runway from the plane is \(6^\circ\).

1. Explain why the angle of elevation of the plane from the runway is also \(6^\circ\). [1 mark]
2. Find the horizontal distance to the runway. Give your answer in km to 1 decimal place. [3 marks]
3. Find the distance flown along the descent path in km. [2 marks]
4. The plane flies at 250 km/h. How many minutes does the descent take? [2 marks]

Priya walks 8 km north and then 6 km east.

1. How far is she from the start in a straight line? [2 marks]
2. Find the bearing of her position from the start, to the nearest degree. [3 marks]
3. What is the bearing of the start from her? [1 mark]
4. She walks back directly at 4 km/h. How long does she take? [2 marks]

A roof is an isosceles triangle with a horizontal span of 8 m and a height of 2.5 m at the centre.

1. Find the length of each sloping side to 2 decimal places. [3 marks]
2. Find the angle each side makes with the horizontal. [3 marks]
3. The roof is covered with tiles costing £25 per m² and the building is 12 m long. Find the cost for both sloping sides. [3 marks]

A television screen has a diagonal of 55 inches. The width and height are in the ratio \(16:9\).

1. Let the width be \(16k\) and the height \(9k\). Show that \(k^2=\dfrac{55^2}{337}\). [3 marks]
2. Find the width and height to 1 decimal place. [3 marks]
3. Find the area of the screen in square inches (nearest whole number). [2 marks]

A road rises 1 m for every 8 m travelled horizontally.

1. Find the angle the road makes with the horizontal to 1 decimal place. [2 marks]
2. The road climbs 25 m in height. How far horizontally does it go? [1 mark]
3. How long is the road itself? [3 marks]
4. Express the gradient as a percentage. [1 mark]

A ski lift cable is 600 m long and rises 180 m.

1. Find the angle of the cable to the horizontal. [2 marks]
2. Find the horizontal distance covered. [2 marks]
3. The lift travels at 2.5 m/s along the cable. How many minutes does a ride take? [2 marks]
4. A second cable is parallel and 100 m long. How high does it rise? [2 marks]

A cuboid measures 8 cm by 6 cm by 5 cm. \(ABCD\) is the base (8 by 6) and \(E\) is above \(A\) with \(AE=5\) cm.

1. Find the diagonal \(AC\) of the base. [2 marks]
2. Find the space diagonal \(EC\) to 2 decimal places. [3 marks]
3. Find the angle \(EC\) makes with the base, to 1 decimal place. [3 marks]
4. Show that \(EC^2=8^2+6^2+5^2\). [2 marks]

A pyramid has a square base of side 10 cm and a vertical height of 12 cm above the centre of the base.

1. Find the length of a half-diagonal of the base. [2 marks]
2. Find the length of a sloping edge to 2 decimal places. [3 marks]
3. Find the angle a sloping edge makes with the base. [3 marks]
4. Find the slant height of a triangular face and its area. [4 marks]

From a point \(A\) on level ground the angle of elevation of the top of a tower is \(25^\circ\). From a point \(B\), 40 m closer to the tower in a straight line, the angle of elevation is \(40^\circ\).

1. Let \(h\) be the height of the tower. Write the distance from \(A\) and from \(B\) to the base in terms of \(h\). [2 marks]
2. Show that \(\dfrac{h}{\tan25^\circ}-\dfrac h{\tan40^\circ}=40\). [2 marks]
3. Solve it to find \(h\) to 1 decimal place. [3 marks]

An equilateral triangle has sides 6 cm.

1. Find the exact height of the triangle. [3 marks]
2. Find the exact area. [2 marks]
3. Show that \(\sin60^\circ=\dfrac{\sqrt3}2\) using your triangle. [2 marks]
4. A right-angled isosceles triangle has equal sides 1. Find the exact value of \(\sin45^\circ\). [3 marks]

A right-angled triangle has sides \(x\) cm, \((x+1)\) cm and \((x+2)\) cm, where \(x+2\) is the hypotenuse.

1. Show that \(x^2-2x-3=0\). [3 marks]
2. Solve the equation. [2 marks]
3. Find the area of the triangle. [2 marks]
4. Find the smallest angle to 1 decimal place. [2 marks]

For whole numbers \(m>n\), a triple is \((m^2-n^2,\ 2mn,\ m^2+n^2)\).

1. Use \(m=5\) and \(n=2\) to find a triple and verify it. [3 marks]
2. Prove that \((m^2-n^2)^2+(2mn)^2=(m^2+n^2)^2\). [4 marks]
3. Find a triple where the hypotenuse is 25. [3 marks]