Many triangles, one connected model
Multi-step problems contain two or more right triangles. Give each triangle its own angle equation, then link them through a shared height, shared base segment, height difference or total distance.
Learning goals
- Decompose a situation into linked right triangles and shared quantities.
- Introduce variables and form a consistent system from multiple observations.
- Account for height offsets and collinear distance relations.
- Solve and validate multi-stage motion, shadow or two-object applications.
Concept lesson
One triangle, one ratio equation
For each observation triangle, identify opposite and adjacent sides and write its own tangent equation before combining anything.
Eliminate a shared unknown
Rearrange each ratio equation to express the shared height or distance, equate the two expressions, solve one variable and substitute back.
Shared-quantity table
Create one row per right triangle recording angle, opposite, adjacent and equation. Add a link column identifying a shared height, a sum or difference of bases, or a known vertical offset. This prevents quantities from different triangles being mixed.
Distance direction from physical layout
If two objects lie on opposite sides of the observer, their separation is the sum of observer-to-object distances. If they lie on the same ray, the separation is the larger distance minus the smaller. Draw the point order before forming the relation.
Two observations of one height
For a tower seen at 60 degrees from a nearer point and 30 degrees from a point 20 m farther away, write h/x=sqrt(3) and h/(x+20)=1/sqrt(3). Equate h expressions, solve x, then find h.
Separate upper height from total height
For a transmission tower on a 20 m building, the 45-degree sight line to the tower base gives horizontal distance, while the 60-degree sight line to the top gives total height. Subtract 20 m only after finding the total.
Motion with unit conversion
Convert speed from km/h to m/s before using elapsed seconds. Use the two angle equations to express the vehicle's distances from the tower, subtract to match distance travelled in the given time, then use uniform speed to obtain the requested time.
One row per triangle
Complete a shared-quantity table for three problems, form separate tangent equations and solve only after the link between triangles is explicit.
Audit height and distance composition
For diagrams with platforms, observers between objects and same-side objects, correct proposed sum/difference relations by writing the physical point order and segment equations.
Speed-distance-time consistency
Convert speeds, calculate travelled distance for an elapsed interval and compare it with the difference between trigonometric position distances before solving a motion question.
Two-object and two-angle synthesis
Model a pole and tower or lighthouse, tower and ship configuration; derive each required height and horizontal distance, then calculate a sloping or top-to-top distance using Pythagoras when requested.
Four-mark modelling checklist
Label every point and segment, justify depression-angle transfer, write one equation per triangle, show height/base sum or difference relations, convert units, answer every requested quantity and attach units.
Multi-step application mastery
Solve one offset-height problem, one opposite-side or same-side two-object problem and one uniform-motion problem using shared-quantity tables, exact ratios, unit conversion and reasonableness checks.