If A (20, 10), B(0, 20) are given, find the coordinates of the points which divide segment AB into five congruent parts
If A (20, 10), B(0, 20) are given, find the coordinates of the points which divide segment AB into five congruent parts
Answer:-
Let the points [Pleft( x_1 , y_1 right), Qleft( x_2 , y_2 right), Rleft( x_3 , y_3 right) text { and } Sleft( x_4 , y_4 right)] be the points which divide the line segment AB into 5 equal parts.
[frac{AP}{PB} = frac{AP}{PQ + QR + RS} = frac{AP}{4AP} = frac{1}{4}]
[x_1 = left( frac{1 times 0 + 4 times 20}{1 + 4} right) = 16]
[ y_1 = left( frac{1 times 20 + 4 times 10}{1 + 4} right) = 12]
[Pleft( x_1 , y_1 right) = left( 16, 12 right)]
[frac{PQ}{QB} = frac{PQ}{QR + RS + SB} = frac{PQ}{PQ + PQ + PQ} = frac{PQ}{3PQ} = frac{1}{3}]
[x_2 = left( frac{1 times 0 + 3 times 16}{1 + 3} right) = 12]
[ y_2 = left( frac{1 times 20 + 3 times 12}{1 + 3} right) = 14]
[Qleft( x_2 , y_2 right) = left( 12, 14 right)]
[frac{QR}{RB} = frac{QR}{RS + SB} = frac{QR}{QR + QR} = frac{QR}{2QR} = frac{1}{2}]
[x_3 = left( frac{1 times 0 + 2 times 12}{1 + 2} right) = 8]
[ y_3 = left( frac{1 times 20 + 2 times 14}{1 + 2} right) = 16]
[Rleft( x_3 , y_3 right) = left( 8, 16 right)]
S is the midpoint of RB so, using the midpoint formula
[x_4 = frac{8 + 0}{2} = 4]
[ y_4 = frac{16 + 20}{2} = 18]
[Sleft( x_4 , y_4 right) = left( 4, 18 right)]
So, the points
[Pleft( x_1 , y_1 right) = left( 16, 12 right)]
[ Qleft( x_2 , y_2 right) = left( 12, 14 right)]
[Rleft( x_3 , y_3 right) = left( 8, 16 right)]
[Sleft( x_4 , y_4 right) = left( 4, 18 right)]
Chapter 5. Co-ordinate Geometry – Practice Set 5.2 (Page 115)