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Ecolint Campus des NationsMathématiques
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Problem-solving Pack

Ecolint Campus des NationsMathematics
Year 9 · 9.1 Surds and Pythagoras

Problem-solving Pack

Name: _________________________________
Date: _________________ Class: ___________

These problems are designed to challenge you. Read each question carefully. Show all your reasoning — a correct answer without working receives no credit.

1Problem 1 of 12
Leaning ladder. A ladder of length 10 m leans against a vertical wall. The foot of the ladder is 3 m from the base.

(a) Sketch the situation.
(b) Find the height reached, to 2 d.p.
(c) If the wall is 12 m tall, find the distance from the top of the ladder to the top of the wall.

Working space

2Problem 2 of 12
Pythagorean triples. Pythagorean triples are sets of three positive integers (a,b,c)(a, b, c) satisfying a2+b2=c2a^2 + b^2 = c^2.

(a) Verify (3,4,5)(3, 4, 5), (5,12,13)(5, 12, 13), (8,15,17)(8, 15, 17) are triples.
(b) Show (7,24,25)(7, 24, 25) is also a triple.
(c) Multiply (3,4,5)(3, 4, 5) by 7 to find another triple.
(d) Show that if (a,b,c)(a, b, c) is a triple, then so is (ka,kb,kc)(ka, kb, kc) for any positive integer kk.

Working space

3Problem 3 of 12
3D Pythagoras. A cuboid has length 4, width 3, height 2 m.

(a) Find the diagonal of the base.
(b) Find the space diagonal (corner to opposite corner).
(c) Find the angle between the space diagonal and the base.

Working space

4Problem 4 of 12
Surds simplification. Simplify each:

(a) 72\sqrt{72}
(b) 12+27\sqrt{12} + \sqrt{27}
(c) 50×2\sqrt{50} \times \sqrt{2}
(d) 82\dfrac{8}{\sqrt{2}} (rationalised)

Working space

5Problem 5 of 12
Diagonal of a cube. A cube has side length aa.

(a) Find the face diagonal in terms of aa.
(b) Find the space diagonal.
(c) For a=5a = 5 cm, compute both diagonals.

Working space

6Problem 6 of 12
The fish tank. A fish tank is 60 × 40 × 30 cm. A diagonal frame is built from corner to opposite corner.

(a) Find the length of the frame.
(b) Find the angle of the frame above the bottom.
(c) Find the lengths needed for two diagonals across each face.

Working space

7Problem 7 of 12
Triangle perimeter with surds. A triangle has vertices A(0,0),B(3,4),C(0,4)A(0, 0), B(3, 4), C(0, 4).

(a) Find each side.
(b) Show one side is a Pythagorean triple-like length.
(c) Find the perimeter.

Working space

8Problem 8 of 12
Diagonal of a rectangular field. A rectangular field 60 m × 80 m has a diagonal path.

(a) Find the path length.
(b) Find the angle the path makes with the longer side.
(c) The field is enlarged by a scale factor of 1.5. Find the new diagonal length.

Working space

9Problem 9 of 12
Surds in algebra. A square has area 5050 cm².

(a) Find the side length as a simplified surd.
(b) Find the diagonal of the square.
(c) Find the perimeter.

Working space

10Problem 10 of 12
Application: angle of elevation. A pole of height 12 m casts a shadow 5 m long.

(a) Find the distance from the tip of the shadow to the top of the pole.
(b) Find the angle of elevation of the sun.
(c) Identify any Pythagorean triple.

Working space

11Problem 11 of 12
Rationalising denominators. Rationalise each:

(a) 52\dfrac{5}{\sqrt{2}}
(b) 13−1\dfrac{1}{\sqrt{3} - 1}
(c) 35+2\dfrac{3}{\sqrt{5} + \sqrt{2}}

Working space

12Problem 12 of 12
Pythagoras in space. A box has dimensions a,b,ca, b, c. The space diagonal dd satisfies d2=a2+b2+c2d^2 = a^2 + b^2 + c^2.

(a) Express dd in surd form for a=1,b=2,c=2a = 1, b = 2, c = 2.
(b) Find the smallest cube containing a stick of length 10 m.
(c) Investigate: can the space diagonal of a cube equal twice the side length?

Working space