Converse of Pythagoras' Theorem

1Summary 重點摘要

Learn how to use Pythagoras' theorem in reverse to tell if a triangle is right triangle.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: The statement of the converseThe converse of Pythagoras' theorem runs the theorem in reverse: if the three sides a, b, c of a triangle satisfy the condition that the square of the longest side equals the sum of the squares of the other two, that is c² = a² + b² with c the longest side, then the triangle must be right-angled, with the right angle opposite the longest side. For sides 3, 4, 5, since 5² = 3² + 4², the triangle is right-angled. Why useful? Real problems often give side lengths but no angles. A common mistake is testing with the shortest side as c, which inverts the whole check.
2. Step 2: The procedureApply the converse in four steps: first, find the longest of the three sides and call it c; second, compute c², the square of the longest side; third, compute a² + b², the sum of squares of the other two sides; fourth, compare the two numbers, equal means right-angled, unequal means not. For sides 8, 15, 17: the longest side is 17, 17² = 289, and 8² + 15² = 64 + 225 = 289; equal, so the triangle is right-angled. A common mistake is skipping the first step and treating an arbitrary side as c, making the test meaningless.
3. Step 3: An example where they are equalTest sides 3, 4, 5 completely: the longest side is 5, and 5² = 25; the sum of squares of the other two is 3² + 4² = 9 + 16 = 25. The two agree, so by the converse the triangle is right-angled, with the right angle opposite the side of length 5. Likewise for 7, 24, 25: 25² = 625 and 7² + 24² = 49 + 576 = 625. Why do examples favour such numbers? They are Pythagorean triples with tidy arithmetic. A common mistake is an arithmetic slip in squaring, for example writing 24² = 48; recheck the squared values once.
4. Step 4: An example where they differIf the square of the longest side does not equal the sum of squares of the other two, the triangle is not right-angled. For sides 2, 3, 4: the longest side 4 gives 4² = 16, while 2² + 3² = 4 + 9 = 13. Since 16 does not equal 13, the triangle is not right-angled; moreover 16 is greater than 13, meaning the longest side is too long, so the triangle is obtuse. Why note the direction? More than 13 indicates an obtuse triangle, less than 13 an acute one, which classifies the triangle at the same time. A common mistake is writing the conclusion without the numerical comparison, losing all method marks.
5. Step 5: Practical applicationsThe converse has wide practical use: a carpenter wanting a right-angled corner can measure 30 and 40 centimetres along the two edges and then the diagonal; if it measures exactly 50 centimetres, since 30² + 40² = 50², the converse guarantees a right angle. This is a standard check in building work. Similarly, given the three distances between locations on a map, the converse decides whether the three form a right-angled arrangement. Why reliable? Lengths are easy to measure accurately while angles are not. A common mistake is imprecise measurement, where errors make equal values appear unequal.
6. Step 6: Points to rememberThree points matter when applying the converse: first, the longest side must sit alone on one side of the equation, because the hypotenuse of a right triangle is always the longest side; second, the converse decides only whether the triangle is right-angled, not the size of the acute angles; third, confirm the three lengths belong to the same triangle. For sides 9, 12, 15: 15² = 225 and 9² + 12² = 81 + 144 = 225, equal, so right-angled. A common mistake is testing with the longer of the two short sides as c, say 12, which can never balance and yields a wrong conclusion.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Converse: reverse check right triangle
• First find longest side
• Longest side squared = sum of other two squares?
• Equal = right triangle
• Not equal = not