Pythagoras' Theorem

The relationship between the three sides of a right-angled triangle — how to find a missing side, and how to use it backwards to prove an angle is a right angle.

หลักสูตรแกนกลาง / สสวท.: การวัดและเรขาคณิต (Geometry and Measure)ตรงกับ ม.2 บทที่ 1 ทฤษฎีบทพีทาโกรัส
Common Core (US): 8.G
Cambridge: Lower Secondary Stage 8 — Geometry and Measure: Pythagoras' theorem

8.1 The theorem

นิยาม

Hypotenuse

The side opposite the right angle. It is always the longest side, and identifying it before doing anything else is what keeps the arithmetic the right way round.

Pythagoras' theorem

a^{2} + b^{2} = c^{2}

$c$ is the hypotenuse; $a$ and $b$ are the two shorter sides. It applies to right-angled triangles and to nothing else.

ทำไมถึงเป็นแบบนั้น

The theorem is about AREA, not length. Draw a square on each side and the two smaller squares together have exactly the area of the large one — which is why the sides are squared rather than added.

ตัวอย่าง 1

A right-angled triangle has shorter sides $6$ cm and $8$ cm. Find the hypotenuse.

วิธีทำ

1. $c^{2} = 6^{2} + 8^{2} = 36 + 64 = 100$
2. $c = 100$

ตอบ $10$ cm

ตัวอย่าง 2

A ladder $13$ m long leans against a wall with its foot $5$ m out. How high up the wall does it reach?

วิธีทำ

1.The ladder is the hypotenuse, so $c = 13$ and one short side is $5$
2. $a^{2} = 1 3^{2} - 5^{2} = 169 - 25 = 144$

ตอบ $12$ m

ข้อควรระวัง

Finding a shorter side means SUBTRACTING, not adding. If you end up taking the square root of a negative number, you have put the hypotenuse in the wrong place — go back to the diagram.

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