Secondary 2 Mathematics Geometry Trigonometry Quiz

Secondary 2 Mathematics Quiz - Geometry Trigonometry

Answer Key

Section A

1. [2]

Using Pythagoras' theorem: $AC^2 = AB^2 + BC^2 = 8^2 + 15^2 = 64 + 225 = 289$ $AC = \sqrt{289} = 17 \text{ cm}$

Answer: $AC = 17$ cm

Marking: M1 for correct use of Pythagoras' theorem, A1 for correct answer.

2. [2]

Using Pythagoras' theorem: $\text{Ladder}^2 = 5^2 + 12^2 = 25 + 144 = 169$ $\text{Ladder} = \sqrt{169} = 13 \text{ m}$

Answer: Length of ladder = 13 m

Marking: M1 for correct use of Pythagoras' theorem, A1 for correct answer.

3. [2]

First find $QR$ using Pythagoras' theorem: $QR^2 = PR^2 - PQ^2 = 26^2 - 10^2 = 676 - 100 = 576$ $QR = \sqrt{576} = 24 \text{ cm}$

$\sin \angle QPR = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{QR}{PR} = \frac{24}{26} = \frac{12}{13}$

Answer: $\sin \angle QPR = \frac{12}{13}$

Marking: M1 for finding $QR$ or setting up ratio correctly, A1 for correct answer.

4. [2]

Check using the converse of Pythagoras' theorem: $AB^2 + BC^2 = 6^2 + 8^2 = 36 + 64 = 100 = 10^2 = CA^2$

Since $AB^2 + BC^2 = CA^2$, by the converse of Pythagoras' theorem, the triangle is right-angled at $B$ (the angle opposite the longest side $CA$).

Answer: Triangle $ABC$ is right-angled at $B$.

Marking: M1 for checking $6^2 + 8^2 = 10^2$, A1 for correct conclusion with right angle identified.

5. [2]

First find $XZ$ using Pythagoras' theorem: $XZ^2 = XY^2 + YZ^2 = 7^2 + 24^2 = 49 + 576 = 625$ $XZ = \sqrt{625} = 25 \text{ cm}$

$\tan \angle XZY = \frac{\text{opposite}}{\text{adjacent}} = \frac{YZ}{XY} = \frac{24}{7}$

Answer: $\tan \angle XZY = \frac{24}{7}$

Marking: M1 for correct identification of opposite and adjacent sides relative to $\angle XZY$, A1 for correct answer.

6. [2]

Using similar triangles (same sun angle): $\frac{\text{height of pole}}{15} = \frac{2}{3}$ $\text{height of pole} = \frac{2}{3} \times 15 = 10 \text{ m}$

Answer: Height of the pole = 10 m

Marking: M1 for setting up correct proportion, A1 for correct answer.

7. [2]

Check: $9^2 + 12^2 = 81 + 144 = 225 = 15^2$, so the triangle is right-angled at $E$.

$\text{Area} = \frac{1}{2} \times 9 \times 12 = 54 \text{ cm}^2$

Answer: Area = 54 cm²

Marking: M1 for identifying right angle or using $\frac{1}{2}bh$, A1 for correct answer.

8. [2]

Since $\triangle ABC \sim \triangle PQR$, corresponding sides are in the same ratio: $\frac{PQ}{AB} = \frac{QR}{BC}$ $\frac{10}{4} = \frac{QR}{6}$ $QR = \frac{10}{4} \times 6 = 15 \text{ cm}$

Answer: $QR = 15$ cm

Marking: M1 for setting up correct ratio of corresponding sides, A1 for correct answer.

9. [2]

$\tan \angle LNM = \frac{LM}{MN} = \frac{5}{12}$

Since $LM = 15$ cm: $\frac{15}{MN} = \frac{5}{12}$ $MN = \frac{15 \times 12}{5} = 36 \text{ cm}$

Answer: $MN = 36$ cm

Marking: M1 for setting up equation using $\tan$, A1 for correct answer.

10. [2]

The ship forms a right-angled triangle with legs 30 km (east) and 40 km (north).

$\tan \theta = \frac{40}{30} = \frac{4}{3}$ $\theta = \tan^{-1}\left(\frac{4}{3}\right) \approx 53.1^\circ$

Bearing is measured clockwise from north: $\text{Bearing} = 90^\circ - 53.1^\circ = 36.9^\circ$

Answer: Bearing of $C$ from $A$ = $036.9^\circ$ (or $037^\circ$ to nearest degree)

Marking: M1 for correct trigonometric setup or diagram, A1 for correct bearing to 3 s.f. or 1 d.p.

Section B

11. [4]

(a) [2] $BC^2 = AB^2 + AC^2 = 9^2 + 12^2 = 81 + 144 = 225$ $BC = \sqrt{225} = 15 \text{ cm}$

Answer: $BC = 15$ cm

(b) [2] $\cos \angle ACB = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{AC}{BC} = \frac{12}{15} = \frac{4}{5}$

Answer: $\cos \angle ACB = \frac{4}{5}$ (or 0.8)

Marking: (a) M1 for Pythagoras, A1 for answer. (b) M1 for correct ratio, A1 for answer.

12. [3]

The ratio of areas of similar triangles equals the square of the ratio of corresponding sides: $\frac{\text{Area of } \triangle DEF}{\text{Area of } \triangle ABC} = \frac{64}{36} = \frac{16}{9}$

Ratio of corresponding sides: $\frac{DE}{AB} = \sqrt{\frac{16}{9}} = \frac{4}{3}$

$DE = \frac{4}{3} \times 9 = 12 \text{ cm}$

Answer: $DE = 12$ cm

Marking: M1 for finding area ratio, M1 for taking square root to get side ratio, A1 for correct answer.

13. [4]

(a) [2] $PR^2 = PQ^2 + QR^2 = 5^2 + 12^2 = 25 + 144 = 169$ $PR = \sqrt{169} = 13 \text{ cm}$

Answer: $PR = 13$ cm

(b) [2] $\sin \angle QPR = \frac{QR}{PR} = \frac{12}{13}$ $\cos \angle QPR = \frac{PQ}{PR} = \frac{5}{13}$ $\sin \angle QPR + \cos \angle QPR = \frac{12}{13} + \frac{5}{13} = \frac{17}{13}$

Answer: $\sin \angle QPR + \cos \angle QPR = \frac{17}{13}$

Marking: (a) M1 for Pythagoras, A1 for answer. (b) M1 for finding both trig ratios, A1 for correct sum.

14. [4]

(a) [2] $\tan 35^\circ = \frac{ST}{25}$ $ST = 25 \times \tan 35^\circ = 25 \times 0.7002 \approx 17.5 \text{ m}$

Answer: Height of tower $ST = 17.5$ m (to 3 s.f.)

(b) [2] $\tan \theta = \frac{17.5}{10} = 1.75$ $\theta = \tan^{-1}(1.75) \approx 60.3^\circ$

Answer: Angle of elevation from $V = 60.3^\circ$ (to 1 d.p.)

Marking: (a) M1 for correct trig equation, A1 for answer. (b) M1 for using height from (a) and correct trig setup, A1 for answer.

15. [4]

(a) [2] $AC = \sqrt{(7-1)^2 + (10-2)^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \text{ units}$

Answer: $AC = 10$ units

(b) [2] $AB$ is horizontal (same $y$-coordinate): length $AB = 7 - 1 = 6$ units $BC$ is vertical (same $x$-coordinate): length $BC = 10 - 2 = 8$ units

Since $AB$ is horizontal and $BC$ is vertical, $\angle ABC = 90^\circ$.

$\text{Area} = \frac{1}{2} \times AB \times BC = \frac{1}{2} \times 6 \times 8 = 24 \text{ square units}$

Answer: Right angle at $B$; Area = 24 square units

Marking: (a) M1 for distance formula, A1 for answer. (b) M1 for showing perpendicular sides, A1 for area.

16. [5]

(a) [2] Check if triangle is right-angled: $13^2 + 14^2 = 169 + 196 = 365 \neq 225 = 15^2$ $13^2 + 15^2 = 169 + 225 = 394 \neq 196 = 14^2$ $14^2 + 15^2 = 196 + 225 = 421 \neq 169 = 13^2$

Since none of these satisfy Pythagoras' theorem, triangle $XYZ$ is not right-angled.

(b) [3]

Using the cosine rule to find $\angle XYZ$: $\cos \angle XYZ = \frac{XY^2 + YZ^2 - XZ^2}{2 \times XY \times YZ} = \frac{13^2 + 14^2 - 15^2}{2 \times 13 \times 14} = \frac{169 + 196 - 225}{364} = \frac{140}{364} = \frac{35}{91}$

$\angle XYZ = \cos^{-1}\left(\frac{35}{91}\right) \approx 67.38^\circ$

$\sin \angle XYZ = \sin 67.38^\circ \approx 0.9231$

$\text{Area} = \frac{1}{2} \times XY \times YZ \times \sin \angle XYZ = \frac{1}{2} \times 13 \times 14 \times 0.9231 \approx 84.0 \text{ cm}^2$

Answer: Area $\approx 84.0$ cm² (to 3 s.f.)

Marking: (a) M1 for checking Pythagoras, A1 for correct conclusion. (b) M1 for cosine rule, M1 for finding $\sin$ of angle, A1 for area.

17. [4]

(a) [2]

Check: $8^2 + 15^2 = 64 + 225 = 289 = 17^2$, so $\triangle PQR$ is right-angled at $Q$.

The shortest side of $\triangle PQR$ is $PQ = 8$ cm (opposite the smallest angle).

Since $\triangle PQR \sim \triangle STU$, the shortest side of $\triangle STU$ corresponds to $PQ$, which is $ST$.

Answer: $ST = 24$ cm is the shortest side of $\triangle STU$, corresponding to $PQ$.

(b) [2]

Scale factor $= \frac{24}{8} = 3$

Perimeter of $\triangle PQR = 8 + 15 + 17 = 40$ cm

Perimeter of $\triangle STU = 40 \times 3 = 120$ cm

Answer: Perimeter of $\triangle STU = 120$ cm

Marking: (a) M1 for identifying shortest side, A1 for correct correspondence. (b) M1 for scale factor, A1 for perimeter.

Section C

18. [7]

(a) [2] $AC = \sqrt{20^2 + 12^2} = \sqrt{400 + 144} = \sqrt{544} \approx 23.3 \text{ m}$

Answer: $AC \approx 23.3$ m (to 3 s.f.)

(b) [2] $\tan \angle CAB = \frac{12}{20} = 0.6$ $\angle CAB = \tan^{-1}(0.6) \approx 31.0^\circ$

Answer: Angle $\approx 31.0^\circ$

(c) [3]

Diagonal of $ABCD = \sqrt{544}$ m. Diagonal of $EFGH = 39$ m.

Scale factor $= \frac{39}{\sqrt{544}}$

Area of $ABCD = 20 \times 12 = 240$ m²

Area of $EFGH = 240 \times \left(\frac{39}{\sqrt{544}}\right)^2 = 240 \times \frac{1521}{544} = \frac{365040}{544} = 671.0$ m²

Answer: Area of $EFGH = 671$ m² (to 3 s.f.)

Marking: (a) M1 for Pythagoras, A1 for answer. (b) M1 for correct trig ratio, A1 for angle. (c) M1 for scale factor from diagonals, M1 for area scale factor, A1 for answer.

19. [6]

(a) [3]

The angle of depression from the cliff top to the boat equals the angle of elevation from the boat to the cliff top ($28^\circ$).

$\tan 28^\circ = \frac{80}{d_1}$ $d_1 = \frac{80}{\tan 28^\circ} = \frac{80}{0.5317} \approx 150.5 \text{ m}$

Answer: Horizontal distance $\approx 150$ m (to 3 s.f.)

(b) [3]

After sailing further, let the new horizontal distance be $d_2$: $\tan 18^\circ = \frac{80}{d_2}$ $d_2 = \frac{80}{\tan 18^\circ} = \frac{80}{0.3249} \approx 246.2 \text{ m}$

Additional distance $= d_2 - d_1 = 246.2 - 150.5 = 95.7$ m

Answer: Additional distance $\approx 95.7$ m (to 3 s.f.)

Marking: (a) M1 for correct diagram/trig setup, M1 for correct equation, A1 for answer. (b) M1 for new distance, M1 for subtraction, A1 for answer.

20. [6]

(a) [3]

Using the cosine rule: $AC^2 = AB^2 + BC^2 - 2 \times AB \times BC \times \cos \angle ABC$ $AC^2 = 10^2 + 18^2 - 2 \times 10 \times 18 \times \cos 52^\circ$ $AC^2 = 100 + 324 - 360 \times 0.6157$ $AC^2 = 424 - 221.6 = 202.4$ $AC = \sqrt{202.4} \approx 14.2 \text{ cm}$

Answer: $AC \approx 14.2$ cm (to 3 s.f.)

(b) [2] $\text{Area} = \frac{1}{2} \times AB \times BC \times \sin \angle ABC = \frac{1}{2} \times 10 \times 18 \times \sin 52^\circ$ $= 90 \times 0.7880 \approx 70.9 \text{ cm}^2$

Answer: Area $\approx 70.9$ cm² (to 3 s.f.)

(c) [1]

Since $\triangle PQR \cong \triangle ABC$: $PQ = AB = 10 \text{ cm}, \quad QR = BC = 18 \text{ cm}, \quad PR = AC \approx 14.2 \text{ cm}$

Answer: $PQ = 10$ cm, $QR = 18$ cm, $PR \approx 14.2$ cm

Marking: (a) M1 for cosine rule formula, M1 for substitution, A1 for answer. (b) M1 for area formula, A1 for answer. (c) B1 for all three correct.

Total: 40 marks