A 5 m ladder has its base 3 m from a vertical wall
- With level ground, the ladder is the hypotenuse.
- Height = √(5² − 3²) = √16 = 4 m.
- The model gives a 4 m contact height.
Distance & real-world models
Choose the right triangle hidden in the situation.
Draw a diagram before choosing an operation
Translate perpendicular changes and map scales into a justified straight-line distance.
A rectangle diagonal, a taut cable or a direct route across perpendicular paths can form a hypotenuse. Mark the right angle and label known lengths with units. Use Pythagoras only when perpendicularity is given or justified. A sloping surface is not automatically a right triangle.
Explore the relationship
Predict first. Change one value, then explain the result. Diagrams show the entered dimensions; displayed decimals are rounded.
Separate direct distance from travel distance
An invented route goes 6 km east and 8 km north on perpendicular paths. The travelled distance is 14 km. The straight-line distance is √(36 + 64) = 10 km. The direct route need not be an available road. Similarly, coordinate differences give horizontal and vertical lengths; use their magnitudes as leg lengths.
Interpret the length you calculated
A model needs assumptions. For a ladder, assume a straight ladder, vertical wall and level ground; ladder length is the hypotenuse, not the height reached. Use compatible units, keep intermediate precision and state a contextual answer. A geometric calculation alone does not establish that equipment is safe to use.
Worked examples
Find the distance from (−2, 1) to (4, 9)
- Horizontal change = 4 − (−2) = 6 units.
- Vertical change = 9 − 1 = 8 units.
- Direct distance = √(6² + 8²) = 10 units.
Guided Practice
A rectangular field measures 30 m by 40 m. Compare a diagonal crossing with two adjacent sides.
A first hint
The two side lengths are legs of a right triangle.
Compare your reasoning
Diagonal = 50 m. Two adjacent sides total 70 m, so the direct crossing is 20 m shorter.
Practice
Fresh questions · useful feedback
Build fluency, one decision at a time.
Work through eight questions. Two independent correct answers introduce a harder version; supported answers bring the next question back to the foundation. You can use hints at any time.
0 of 8 questions completed
This round saves on this device. It is practice, not a school mark.
Connections
Invent a situation in which the calculated straight-line distance is accurate but cannot be used as the actual travel route. State the model assumptions.
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