Coordinate Geometry

Gradient, midpoint and length from coordinates, and the equations of lines that are parallel or perpendicular to a given one.

หลักสูตรแกนกลาง / สสวท.: การวัดและเรขาคณิต (Geometry and Measure)ตรงกับ ม.5 เรขาคณิตวิเคราะห์
Common Core (US): G-GPE, 8.EE
Cambridge: IGCSE 0580 — Topic 3: Coordinate geometry (Core and Extended)

6.1 Gradient, midpoint and length

Three formulae from two points

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}} M = (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2}) d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

The length formula is Pythagoras on the triangle whose legs are the differences in $x$ and $y$ — not a separate result.

ตัวอย่าง

For $A (- 2, 3)$ and $B (6, 9)$ , find the gradient, midpoint and length of $A B$ .

วิธีทำ

1. $m = \frac{9 - 3}{6 - ( - 2 )} = \frac{6}{8} = \frac{3}{4}$
2. $M = (\frac{- 2 + 6}{2}, \frac{3 + 9}{2}) = (2, 6)$
3. $d = 8^{2} + 6^{2} = 100$

ตอบ $m = \frac{3}{4}$ , $M = (2, 6)$ , $A B = 10$

ข้อควรระวัง

Subtract in the SAME order top and bottom. Doing $y_{2} - y_{1}$ over $x_{1} - x_{2}$ gives the right size with the wrong sign, and nothing about the number looks suspicious.

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