Pack B — Fluency

Mathematics

Year 9 · 9.1 Surds and Pythagoras

Pack B · Fluency

Name: _________________________________

Date: _________________ Class: ___________

Answer all questions. Show your working. Questions are grouped by challenge level.

BronzeQuestions 1–10

1.
A right-angled triangle has the two shorter sides of length 6 cm and 8 cm. Find the hypotenuse.
2.
Simplify $\sqrt{{n}}$ fully.
3.
Simplify $\sqrt{{n}}$ fully.
4.
A triangle has sides 7, 24, 25 cm. Use Pythagoras' converse to decide whether it is right-angled.
5.
Simplify $\sqrt{{a}} \times \sqrt{{b}}$ .
6.
Identify which numbers are rational vs irrational: $5, \sqrt{2}, \pi, 0.\overline{3}, \sqrt{4}$ .
7.
Estimate $\sqrt{50}$ to 1 d.p.
8.
Find the hypotenuse of a right-angled triangle with legs 9 and 12 cm.
9.
Add: $\sqrt{50} + \sqrt{18}$ (give simplified form).
10.
A right-angled triangle has hypotenuse 13 cm and one shorter side 5 cm. Find the other side.

SilverQuestions 11–20

11.
Pythagoras with surds: legs $2\sqrt{3}$ and 3. Find the hypotenuse.
12.
Find the distance between $(1, 2)$ and $(4, 6)$ . Give answer as a surd in simplest form.
13.
Simplify $\sqrt{8} + \sqrt{32} - \sqrt{18}$ .
14.
Find the hypotenuse of a right-angled triangle with legs 5 and 10 cm. Give answer as a simplified surd.
15.
Simplify $\sqrt{75} \div \sqrt{3}$ .
16.
A right-angled triangle has hypotenuse $\sqrt{50}$ and one leg $\sqrt{18}$ . Find the other leg.
17.
Find the distance between $(-2, 1)$ and $(3, 13)$ .
18.
Find the perimeter of a right-angled triangle with legs 6 and 8 cm.
19.
Square the surd: $(2 + \sqrt{3})^2$ . Expand.
20.
Rationalise the denominator: $\dfrac{6}{\sqrt{3}}$ .

GoldQuestions 21–30

21.
A right-angled triangle has shorter sides 5 and 10. Hypotenuse as a simplified surd.
22.
Find the length of the space diagonal of a cuboid with sides 8, 6, 4. Use 3D Pythagoras.
23.
Pythagoras in 3D: cuboid 12 × 9 × 6. Find the space diagonal.
24.
A square has diagonal 10 cm. Find its side length and area.
25.
A ladder of length 10 m leans against a wall. Foot 3 m from the base. Find the height reached.
26.
Simplify $(3 + \sqrt{2})(2 - \sqrt{2})$ .
27.
The area of a square is 12 cm². Find the side as a simplified surd.
28.
Find the length of the diagonal of a rectangle 6 cm by 8 cm.
29.
Show that the triangle with sides 5, 5, $5\sqrt{2}$ is right-angled.
30.
A ladder of length 8 m makes a 60° angle with the ground. Find the height up the wall (using trig).

PlatinumQuestions 31–40

31.
A cuboid has length 8 cm, width 6 cm, height 4 cm. Find the angle between the space diagonal and the base, to 1 d.p.
32.
Make $a$ the subject of $c = \sqrt{a^2 + b^2}$ .
33.
A cliff of height $h$ m stands on level ground. From point A 28 m from the base, the line of sight to the top is 52 m. From B further away, sight to top is 60 m. Find $h$ and B's distance.
34.
Apply Pythagoras to find a diagonal of a square pyramid: base side 8, apex height 12 above the centre. Find the slant distance from a base vertex to the apex.
35.
Rationalise: $\dfrac{1}{1 + \sqrt{2}}$ .
36.
An isosceles triangle has equal sides 13 cm and base 10 cm. Find (a) the perpendicular height, (b) the area.
37.
Find the area of an equilateral triangle of side $s = 10$ cm.
38.
A right-angled triangle has legs $a$ and $b$ with $a + b = 14$ and $a^2 + b^2 = 100$ . Find $a$ and $b$ .
39.
Two ladders lean against opposite walls of an alley. One reaches 6 m up the left wall; the other 8 m up the right wall. The alley is 5 m wide. They cross at some height $h$ . Find $h$ (use similar triangles).
40.
A triangle has sides $\sqrt{8}$ , $\sqrt{18}$ , $\sqrt{32}$ . Show it is right-angled.