Criterion CD Assessment — 14 April
A formative assessment covering all five assessment topics. Answer each question and study the worked solution before moving on.
TOPIC 01
Finding missing sides in right-angled triangles using a² + b² = c²
In any right-angled triangle, the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the other two sides.
Where c is always the hypotenuse. To find a missing shorter side: a² = c² − b²
Right-angled triangle
Area visualisation
Adjust the two shorter sides and watch the hypotenuse update in real time.
Apply Pythagoras: a² + b² = c²
5² + 12² = c² → 25 + 144 = c² → 169 = c²
c = √169 = 13 cm
This is a classic 5-12-13 Pythagorean triple — worth memorising.
Finding a shorter side: rearrange to a² = c² − b²
a² = 25² − 24² → a² = 625 − 576 = 49
a = √49 = 7 cm
7-24-25 is another Pythagorean triple.
Check: 9² + 40² = 81 + 1600 = 1681
41² = 1681 — these are equal, confirming it is right-angled.
The converse of Pythagoras' theorem states: if a² + b² = c², the triangle is right-angled.
The diagonal forms the hypotenuse of a right triangle with legs 9 and 12.
d² = 9² + 12² = 81 + 144 = 225
d = √225 = 15.0 cm
The height bisects the base, so you have a right triangle with hypotenuse 10 cm and base 4 cm.
h² + 4² = 10² → h² = 100 − 16 = 84
h = √84 ≈ 9.2 cm
East and North are perpendicular, so this is a right triangle.
d² = 8² + 6² = 64 + 36 = 100 → d = 10 km
This is the 6-8-10 Pythagorean triple (a scaled 3-4-5).
11² = 121, 60² = 3600, so 11² + 60² = 3721
61² = 3721 — they match, confirming a right angle.
First find hypotenuse: c = √(5² + 12²) = √169 = 13 cm
Perimeter = 5 + 12 + 13 = 30 cm
d² = 2.4² + 3.2² = 5.76 + 10.24 = 16
d = √16 = 4.0 m
Use the distance formula (derived from Pythagoras): d = √((x₂−x₁)² + (y₂−y₁)²)
d = √(6² + 8²) = √(36 + 64) = √100 = 10
d² = 40² + 30² = 1600 + 900 = 2500
d = √2500 = 50 inches — a 50-inch TV!
XZ is the hypotenuse since the right angle is at Y.
XZ² = 15² + 20² = 225 + 400 = 625
XZ = √625 = 25.0 cm
The ladder is the hypotenuse (10 m), base leg is 4 m.
h² = 10² − 4² = 100 − 16 = 84
h = √84 ≈ 9.2 m
Check D: 6² + 7² = 36 + 49 = 85, but 9² = 81 ≠ 85. Not a triple.
A: 9+16=25 ✓ | B: 25+144=169 ✓ | C: 64+225=289 ✓
If side = s, then diagonal d = s√2, so 10 = s√2
s = 10/√2 = 10√2/2 ≈ 7.07 cm
Alternatively: s² + s² = 10² → 2s² = 100 → s = √50 ≈ 7.07
Area = ½ × base × height → 60 = ½ × 10 × b → b = 12 cm
c² = 10² + 12² = 100 + 144 = 244
c = √244 ≈ 15.6 cm — accept ≈15.6 to 16.0
The diagonals of a rhombus bisect each other at right angles. Half-diagonals: 8 cm and 6 cm.
side² = 8² + 6² = 64 + 36 = 100 → side = 10 cm
d = √((5−1)² + (5−2)²) = √(16 + 9) = √25 = 5.000
Using a² = c² − b²: a² = (√97)² − (√3)² = 97 − 3 = 94
a = √94 cm
The height bisects the base. Right triangle: hypotenuse = 6, base = 3.
h² = 6² − 3² = 36 − 9 = 27 → h = √27 ≈ 5.20 cm
TOPICS 02 – 05
SOH CAH TOA, angles of elevation & depression, and bearings
In a right-angled triangle, label sides relative to the angle θ:
To find a missing side: rearrange the trig ratio, e.g. Opp = Hyp × sin θ
To find a missing angle: use the inverse trig function, e.g. θ = sin⁻¹(Opp/Hyp)
Angle of elevation — measured upward from horizontal to a higher object.
Angle of depression — measured downward from horizontal to a lower object.
Bearings — measured clockwise from North (always 3 digits, e.g. 045°, 270°).
Adjust the angle to see how sin, cos, and tan change.
We need Opposite, given Hypotenuse and angle → use SOH.
sin 30° = Opp / Hyp → Opp = 15 × sin 30° = 15 × 0.5 = 7.5 cm
Adjacent and Hypotenuse → use CAH.
cos 40° = Adj / Hyp → Hyp = 8 / cos 40° = 8 / 0.766 ≈ 10.44 cm
We have Opposite and Adjacent → use TOA.
tan θ = 5 / 12 = 0.4167
θ = tan⁻¹(0.4167) ≈ 22.6°
x is the adjacent side, and we know the hypotenuse → use CAH.
cos 50° = Adjacent / Hypotenuse = x / 20
Therefore x = 20 × cos 50° ≈ 12.86 cm
The shadow is the adjacent side, the pole height is opposite, angle = 35°.
tan 35° = h / 12 → h = 12 × tan 35° ≈ 12 × 0.700 ≈ 8.4 m
Angle of depression from horizontal = 20°. The angle in the right triangle is also 20°.
Cliff height = opposite (80 m), distance = adjacent.
tan 20° = 80 / d → d = 80 / tan 20° ≈ 80 / 0.364 ≈ 219.8 m
Opposite = 50 m (height), Adjacent = 40 m.
tan θ = 50 / 40 = 1.25 → θ = tan⁻¹(1.25) ≈ 51.3°
090° is East (30 km), 000° is North (40 km). These are perpendicular.
d² = 30² + 40² = 900 + 1600 = 2500 → d = 50 km
PQ = 26 cm (hypotenuse, opposite R), QR = 10 cm (opposite P → no, it's adjacent to P from R's side).
Relative to angle P: Opposite = QR = 10, Hypotenuse = PQ = 26.
sin P = 10/26 = 0.3846 → P = sin⁻¹(0.3846) ≈ 22.6°
Opposite = 3 m (rise), Adjacent = 15 m (run).
tan θ = 3/15 = 0.2 → θ = tan⁻¹(0.2) ≈ 11.3°
Bearing 045° = NE direction. The East component is opposite to the 45° angle from North.
East = 70 × sin 45° ≈ 70 × 0.7071 ≈ 49.5 km
The angle between the horizontal and the line of sight = 25° (depression = same as elevation from car).
tan 25° = 120/d → d = 120/tan 25° ≈ 120/0.4663 ≈ 257.3 m
Bearings go clockwise from North: N=000°, E=090°, S=180°, W=270°.
Due West = 270°.
Bearing 130° is 130° clockwise from North. Angle from South direction = 130° − 90° = 40° from East.
South component = 200 × cos 40° — wait, easier: angle from North = 130°.
South component = 200 × cos(130° − 90°) = 200 × cos 40°? No. South = 200 × cos(180°−130°) × (−1)
Use: South = 200 × sin(130°−90°) … . Actually: South component = 200 × cos(130°) ×(−1) = −200cos130° = 200 × sin40° ≈ 128.6 km
The back-bearing (return bearing) = original bearing + 180°.
060° + 180° = 240°. If the result exceeds 360°, subtract 360°.
Wire = hypotenuse, pole height = opposite, angle = 55°.
sin 55° = h / 18 → h = 18 × sin 55° ≈ 18 × 0.819 ≈ 14.7 m
Height from eye level: h = 25 × tan 42° ≈ 25 × 0.900 ≈ 22.5 m
Add observer's eye height: Total height = 22.5 + 1.8 = 24.3 m
cos θ = Adj/Hyp = 5/13, so Adj = 5, Hyp = 13.
By Pythagoras: Opp = √(13² − 5²) = √(169 − 25) = √144 = 12
sin θ = Opp/Hyp = 12/13
tan 15° = 500 / d → d = 500 / tan 15° ≈ 500 / 0.2679 ≈ 1866 m
Equal North and East distances → 45° from North, towards East.
Bearing = 045°. (Using tan⁻¹(5/5) = 45° from North, measured clockwise.)
Review any explanations you missed — good luck on 14 April.