ANALYTIC GEOMETRY(Lesson 4)

Math 14 Plane and Analytic Geometry

OBJECTIVES:At the end of the lesson, the student is expected to be

able to:• Determine the coordinates of a point of division of a

line segment.• Define the median of the triangle.

DIVISION OF A LINE SEGMENT

21

1

PP

PPr

222 y,xP

y,xP

111 y,xP 12 y,xN 1y,xM

Internal Point of Division

222 y,xP

y,xP

111 y,xP 12 y,xN 1y,xM

External Point of Division

21

1

PP

PPr

21 P than P from farther always is P point, external For :note

21

1221

21

1221

2

1

2

1

12

1

2

12

1

1

1

21

1

21

1

21

1

rryryr

y and rr

xrxrx then ,

rr

PPPP

If

:Formula eAlternativ

1. r point external for and 21

r

midpoint for 1, r point internal For .yyyy

NPPM

and

;xxxx

NPMP

r;PPPP

But .NP

PMNPMP

PPPP

wherein

,proportion and ratioof use the with computed is y x,P pointof scoordinate the triangle, similar a is figure the Since

Examples:1.The line segment joining (-5, -3) and (3, 4) is to be divided into five equal parts. Find the point of division closest to (-5, -3).2.Find the midpoint of the segment joining (7, -2) and (-3, 5).3.The line segment from (1, 4) to (2, 1) is extended a distance equal to twice its length. Find the terminal point.4.On the line joining (4, -5) to (-4, -2), find the point which is three-seventh the distance from the first to the second point. 5.Find the trisection points of the line joining (-6, 2) and (3, 8).

6. The line segment joining a vertex of a triangle and the midpoint of the opposite side is called the median of the triangle. Given a triangle whose vertices are A(4,-4), B(10, 4) and C(2, 6), find the point on each median that is two-thirds of the distance from the vertex to the midpoint of the opposite side.

REFERENCES

Analytic Geometry, 6th Edition, by Douglas F. RiddleAnalytic Geometry, 7th Edition, by Gordon Fuller/Dalton Tarwater

Analytic Geometry, by Quirino and Mijares