8. Word Problems: Draw the Hidden Triangle

Many right-triangle questions hide the triangle inside a real situation. Your job is to draw it.

Common right-triangle setups:

• A ladder against a wall: ladder = hypotenuse.
• A ramp: ramp surface = hypotenuse; rise and run = legs.
• A rectangle diagonal: diagonal = hypotenuse; length and width = legs.
• A map or grid distance: east/west and north/south moves are legs; straight-line distance is the hypotenuse.

Routine:

• Sketch the right angle.
• Label the legs and hypotenuse.
• Decide whether you are finding the hypotenuse (add) or a leg (subtract).
• Check units and reasonableness. The hypotenuse should be the longest side.

Worked example: a person walks 12 blocks east and 5 blocks north. Straight-line distance:
\[ \sqrt{12^2 + 5^2} = \sqrt{144+25} = \sqrt{169} = 13. \]

Common mistake: adding the walking distances \(12+5=17\). That is the path walked, not the straight-line distance.