Section Formula

Transcript

Section & Mid-point Formula If point P divides AB in ratio m : n Then, coordinates of point P are 𝐏(𝒙, 𝒚)=((𝒎𝒙_𝟐 +𝒏𝒙_𝟏)/(𝒎 + 𝒏),(𝒎𝒙_𝟐 + 𝒏𝒙_𝟏)/(𝒎 + 𝒏)) This is called Section Formula. Now, let’s look at Mid-point Formula Mid-Point FormulaIf point P is the mid-point of AB Then, coordinates of point P are P(𝒙, 𝒚)=((𝒙_𝟏+ 𝒙_𝟐)/𝟐,(𝒚_𝟏 + 𝒚_𝟐)/𝟐) If Ratio is not givenWe assume point P divides AB in ratio k : 1 And, then find k Let’s do an Example Find Ratio in which the line segment joining the points (−3, 10) and (6, −8) is divided by (−1, 6) Let A (−3, 10), B (6, −8) and P (−1, 6) Let P divide AB in ratio k : 1 Therefore, Coordinates of P = ((𝑚𝑥_2 +𝑛𝑥_1)/(𝑚 + 𝑛),(𝑚𝑥_2 + 𝑛𝑥_1)/(𝑚 + 𝑛)) (−1, 6) = ((𝟔𝒌 + (−𝟑))/(𝒌 + 𝟏),(−𝟖𝒌 + 𝟏𝟎)/(𝒌 + 𝟏)) Comparing y-coordinate 𝟔=(−𝟖𝒌 + 𝟏𝟎)/(𝒌 + 𝟏) 6(k+1)=−8𝑘 + 10 6k+6=−8𝑘 + 10 6𝑘+8𝑘=10−6 14𝑘=4 k=4/14 𝐤=𝟐/𝟕 So, P divides AB in ratio 2 : 7