Problems — 9.1 Surds and Pythagoras | Mathematics Department

1

**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.

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2

**Pythagorean triples.** Pythagorean triples are sets of three positive integers $(a, b, c)$ satisfying $a^2 + b^2 = c^2$. (a) Verify $(3, 4, 5)$, $(5, 12, 13)$, $(8, 15, 17)$ are triples. (b) Show $(7, 24, 25)$ is also a triple. (c) Multiply $(3, 4, 5)$ by 7 to find another triple. (d) Show that if $(a, b, c)$ is a triple, then so is $(ka, kb, kc)$ for any positive integer $k$.

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3

**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.

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4

**Surds simplification.** Simplify each: (a) $\sqrt{72}$ (b) $\sqrt{12} + \sqrt{27}$ (c) $\sqrt{50} \times \sqrt{2}$ (d) $\dfrac{8}{\sqrt{2}}$ (rationalised)

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5

**Diagonal of a cube.** A cube has side length $a$. (a) Find the face diagonal in terms of $a$. (b) Find the space diagonal. (c) For $a = 5$ cm, compute both diagonals.

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6

**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.

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7

**Triangle perimeter with surds.** A triangle has vertices $A(0, 0), B(3, 4), C(0, 4)$. (a) Find each side. (b) Show one side is a Pythagorean triple-like length. (c) Find the perimeter.

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8

**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.

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9

**Surds in algebra.** A square has area $50$ cm². (a) Find the side length as a simplified surd. (b) Find the diagonal of the square. (c) Find the perimeter.

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10

**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.

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11

**Rationalising denominators.** Rationalise each: (a) $\dfrac{5}{\sqrt{2}}$ (b) $\dfrac{1}{\sqrt{3} - 1}$ (c) $\dfrac{3}{\sqrt{5} + \sqrt{2}}$

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12

**Pythagoras in space.** A box has dimensions $a, b, c$. The space diagonal $d$ satisfies $d^2 = a^2 + b^2 + c^2$. (a) Express $d$ in surd form for $a = 1, b = 2, c = 2$. (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?

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