Applications of Pythagoras' Theorem

1Summary 重點摘要

Learn how to use Pythagoras' theorem to solve real problems, e.g. ladder, rectangle diagonal.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: The diagonal of a rectangleThe most common application is the diagonal of a rectangle. With length 3 and width 4, cutting along the diagonal gives two congruent right triangles, so the diagonal is the hypotenuse: √(3² + 4²) = √(9+16) = √25 = 5. Recognising the diagonal as the hypotenuse is the key. Memorise the common Pythagorean triples, such as 3, 4, 5 and its multiples like 6, 8, 10, as well as 5, 12, 13; knowing them lets you verify many answers mentally and saves considerable time.
2. Step 2: A ladder against a wallAnother classic type is the ladder against a wall: a 5 m ladder stands with its foot 3 m from the wall; how high is the top? Sketch first — the ladder itself is the hypotenuse, the foot-to-wall distance is one leg, and the wall is the other leg, perpendicular to the ground. Applying the rearranged formula gives height = √(5² - 3²) = √(25-9) = √16 = 4 m. This is the 'given hypotenuse, find a leg' case, so subtract; the worked example, a ladder of 10 with its foot 6 from the wall, works the same way to give 8 m.
3. Step 3: The distance formula between two pointsOn a coordinate plane, the distance between any two points (x₁, y₁) and (x₂, y₂) also follows from Pythagoras' theorem: the horizontal gap is |x₂-x₁|, the vertical gap is |y₂-y₁|, and the segment joining the points is the hypotenuse, giving distance = √[(x₂-x₁)² + (y₂-y₁)²], which is exactly where the distance formula comes from. For example, from (0,0) to (3,4) the distance is √(3² + 4²) = 5. Compute the two coordinate differences first, then square, add and take the square root, keeping the order straight.
4. Step 4: A four-step routine for word problemsWord problems follow a fixed routine: first, draw a diagram to turn the situation into a figure — never skip this; second, locate the right triangle in the diagram and confirm where the right angle is; third, identify the hypotenuse, always the longest side opposite the right angle, with the remaining two as legs; fourth, substitute into the formula — subtract when the hypotenuse is known and a leg is wanted, add when both legs are known. Finally check that the answer is sensible and attach the unit to complete the solution.
5. Step 5: A complete exampleA full example: a rectangular garden measures 12 m by 5 m, and the shortest route from one corner to the opposite corner is the diagonal: √(12² + 5²) = √(144 + 25) = √169 = 13 m. Here 5, 12, 13 is another common Pythagorean triple, so the answer can almost be read off. Note that the question asks for the shortest distance, and a straight line is the shortest route between two points, so the diagonal is correct; if the question instead asked how far one walks along the edges, it would be 12 + 5 = 17 m, and the two answers must not be confused.
6. Step 6: Points to watch in examinationsIn examinations remember three things: draw a diagram and label every side, since matching sides by mental maths alone invites errors; write the answer with its unit, such as 4 m or 13 m, because a bare number loses marks; and if the answer contains a square root with no instruction otherwise, leave it in surd form or round as directed. Verification matters too — substitute back, checking whether 6² + 8² really equals 10² — so you can be sure before moving on and avoid unnecessary losses.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Rectangle diagonal cuts into two right triangles
• Ladder against wall: ladder is hypotenuse
• Distance between two points also uses Pythagoras
• Must draw picture for word problems
• Answer must write units