The triangle and bearings reasoning board makes you decide which model fits before it works anything out. You then enter the known values and get the worked steps, a diagram and some plausibility checks.
It covers Pythagoras, the sine rule, the cosine rule and a two-leg bearings journey. All angles are in degrees.
How do you use it?
- Choose a relationship. The list groups them: right-angled triangle (find a hypotenuse or a leg), sine rule (find a side or an angle), cosine rule (find a side or an angle) and bearings (distance and bearing from the start).
- For a right-angled case, tick the box only if the question states or marks the right angle.
- Enter the known values. The labels change with the relationship you chose, such as Leg a, Angle A or Bearing of B from A.
- Press Work it out. Press Reset to return to the Pythagoras sample.
If you press the button before choosing, the board asks you to choose first. That is deliberate.
How do you read the result?
Steps show the formula, the substitution and the answer line by line. Answer gives the final value. Plausibility checks test whether the result makes sense, for example that the longest side sits opposite the largest angle or that the hypotenuse is the longest side.
The diagram is drawn from your values. For bearings it shows a north line at the start point.
Example walk-through
Pythagoras. The sample is a right-angled triangle with legs 3 and 4. The board writes c² = 3² + 4² = 9 + 16 = 25, so c = √25 = 5. The check confirms the hypotenuse is the longest side.
Ambiguous sine rule. Choose “Find an angle from two sides and an angle opposite one of them” and enter a = 8, b = 10 and A = 40. Then sin B = 10 × sin 40° ÷ 8, which is about 0.8035, so B is about 53.5°. The board notes that B could also be 180° − 53.5° = 126.5°, because 40° + 126.5° is still less than 180°. Two triangles fit.
Bearings. Choose the two-leg journey and use the default values: A to B is 8 units on a bearing of 065°, then B to C is 6 units on a bearing of 140°. East equals distance × sin(bearing) and north equals distance × cos(bearing). The board adds the two legs and finds AC is about 11.2 on a bearing of about 096°. The back bearing from B to A is 065° + 180° = 245°.
What are the assumptions and limits?
- The board does not assume a right angle from how a diagram looks. You must say so.
- It flags the ambiguous case instead of hiding it, but it cannot decide which triangle your question means.
- If sin B would come out above 1, no triangle exists and it says so.
- Three sides that cannot form a triangle are rejected.
- Angles are in degrees, bearings run 0 to 360, and results are rounded for display.
- The diagram is for orientation, not an exam-style construction.
Which lessons explain the ideas behind it?
- Combine Pythagoras with trigonometry covers the right-angled cases and when each tool applies.
- Right-triangle trigonometry practice gives mixed questions.
- Use bearings with north lines consistently explains the north line and the back bearing.
- Non-right triangles and bearings practice covers the sine and cosine rules and bearings together.
The topics sit in right-angled trigonometry and non-right triangles and bearings. If you would like a teacher to watch how you choose a rule, see online one-to-one Mathematics tuition. More tools are in the learning tools directory.