Pythagoras' Theorem

1Summary 重點摘要

Learn Pythagoras' theorem: square of longest side of right triangle equals sum of squares of other two sides.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: The statement of the theoremPythagoras' theorem is among the most famous results in geometry: in a right triangle, the square of the hypotenuse, the longest side opposite the right angle, equals the sum of the squares of the two legs, that is a² + b² = c², where c is the hypotenuse and a, b are the legs. For legs 3 and 4, the hypotenuse is 5 since 3² + 4² = 5². Why only right triangles? The theorem describes exactly the side relation created by the right angle. A common mistake is placing the hypotenuse wrongly, writing a² + c² = b².
2. Step 2: Identifying the hypotenuseBefore applying the theorem, identify the hypotenuse: it is always the side opposite the right angle and the longest side of the triangle. If the right angle is at vertex C, the hypotenuse is AB; if the right-angle mark sits at another vertex, the hypotenuse changes accordingly. Why does this matter? Once c is misplaced, every later addition and subtraction is wrong. Two checks: find the side facing the right-angle mark, or take the longest of the three sides. A common mistake is choosing c without looking at the right angle, especially when it is not in the usual position.
3. Step 3: Finding the hypotenuseGiven the two legs, use c² = a² + b² and take the square root. For a = 3 and b = 4: c² = 9 + 16 = 25, so c = √25 = 5, the classic 3-4-5 triangle; for a = 6 and b = 8, c² = 36 + 64 = 100 and c = 10. Why take only the positive root? A side length is a distance and must be positive. A common mistake is adding the sides without squaring, writing c = 3 + 4 = 7; the correct answer is 5, and the difference is large.
4. Step 4: Finding a legGiven the hypotenuse and one leg, rearrange the formula to b² = c² − a². For c = 5 and a = 3: b² = 25 − 9 = 16, so b = 4; for c = 13 and a = 5, b² = 169 − 25 = 144 and b = 12. Why subtract rather than add? The square of the hypotenuse is the largest, so the known leg's square is deducted from it to find the other leg. A common mistake is reversing the order, giving a² − c², a negative result; a negative value signals that the roles of the sides were mixed up.
5. Step 5: Pythagorean triplesCertain integer triples satisfy a² + b² = c² exactly; they are called Pythagorean triples, and memorising the common ones speeds up calculation greatly: 3-4-5, 5-12-13, 8-15-17, 7-24-25. Multiplying a triple by any whole number gives another, for example twice 3-4-5 is 6-8-10, three times is 9-12-15, five times is 15-20-25. Why are they helpful? Examinations use them constantly; recognising 12 and 13 immediately gives the third side 5. A common mistake is grinding through square roots, or misremembering 5-12-13 as 5-12-15.
6. Step 6: Limits of useTwo restrictions govern the theorem: first, it applies only to right triangles, never to acute or obtuse ones, so a triangle with sides 2, 3, 4, which is not right-angled, cannot be solved with a² + b² = c²; second, take only the positive square root, because lengths cannot be negative. Why stress the first point? Examinations often supply non-right triangles to tempt students into applying the theorem. The converse of the theorem can first confirm whether the angle is right. A common mistake is computing for every triangle; confirm the right angle before answering.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Only for right triangles
• Hypotenuse always longest, opposite right angle
• a² + b² = c²
• After calculating take square root
• Common triples: 3-4-5, 5-12-13