Problems — Geometry | Mathematics Department

1

**Ladder safety.** A 6 m ladder leans against a wall with the foot 2 m from the base of the wall. (a) How high does the ladder reach (to 2 d.p.)? (b) What angle does the ladder make with the ground? (c) Safety advice: the angle should be between 70° and 80°. Is this ladder safely placed? If not, where should the foot be placed?

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2

**Bearings triangle.** A walker sets out from camp $C$ on a bearing of $050°$ for 6 km to point $P$. At $P$, she changes to a bearing of $140°$ and walks 8 km to point $Q$. (a) Show that $\angle CPQ = 90°$. (b) Find the distance $CQ$. (c) Find the bearing of $Q$ from $C$ (to the nearest degree).

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3

**Tower and shadow.** A tower of height $h$ m casts a shadow of length 15 m when the angle of elevation of the sun is $40°$. (a) Find $h$. (b) Later, the shadow has length 25 m. What is the new angle of elevation? (c) When is the sun's angle of elevation $30°$? (Find the shadow length.)

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4

**Non-right-angled triangle.** Triangle $ABC$ has $a = 8$ cm, $b = 11$ cm, $C = 75°$. (a) Use the cosine rule to find $c$. (b) Find $\angle A$ using the sine rule. (c) Find the area of $\triangle ABC$.

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5

**Compound area.** Find the area of an isosceles trapezium with parallel sides 10 cm and 16 cm, and slant sides of length 5 cm.

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6

**Cuboid diagonals.** A cuboid measures 4 cm × 6 cm × 12 cm. (a) Find the length of the space diagonal. (b) Find the angle the space diagonal makes with the base (i.e., the 4 × 6 face).

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7

**Trig identities check.** For a right triangle with sides 3, 4, 5, take the angle $\theta$ opposite the side of length 3. (a) Find $\sin\theta$, $\cos\theta$, $\tan\theta$. (b) Verify $\sin^2\theta + \cos^2\theta = 1$. (c) Verify $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$.

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8

**Two ships.** Two ships leave port at the same time. Ship $A$ sails on bearing $030°$ at 20 km/h; Ship $B$ on bearing $150°$ at 15 km/h. (a) Find the angle between the ships' paths. (b) After 2 hours, how far apart are they (2 d.p.)? (c) What is the bearing of $B$ from $A$ after 2 hours (nearest degree)?

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9

**Pythagorean triples.** A "Pythagorean triple" is a set of three integers $(a, b, c)$ with $a^2 + b^2 = c^2$. (a) Check that $(3, 4, 5)$, $(5, 12, 13)$, and $(8, 15, 17)$ are all Pythagorean triples. (b) Show that if $(a, b, c)$ is a triple, then so is $(ka, kb, kc)$ for any positive integer $k$. (c) Find a triple where $c = 25$ (other than the trivial $(7, 24, 25)$).

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10

**Hexagon.** A regular hexagon is inscribed in a circle of radius 10 cm. (a) Show that each side of the hexagon equals the radius. (b) Find the area of the hexagon (exact, in surd form). (c) Find the area of the circle (in terms of $\pi$) and compare.

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11

**Cone problem.** A cone has a slant height of 13 cm and a base radius of 5 cm. (a) Find the perpendicular height of the cone. (b) Find the volume of the cone in terms of $\pi$. (c) Find the total surface area in terms of $\pi$.

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12

**Architecture.** A roof has a triangular cross-section. The base of the triangle is 8 m wide, and the two slopes meet at an apex 3 m above the base. (a) Find the length of each slope. (b) Find the angle each slope makes with the horizontal (to 1 d.p.). (c) If the roof is 12 m long, find the total area of the two rectangular sloped surfaces.

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