Question

A(5,0) and B(0,8) are two vertices of triangle OAB.
a). What is the equation of the bisector of angle OAB.
b). If E is the point of intersection of this bisector and the line through A and B,find the coordinates of E.
Hence show that OA:OB = AE:EB

Answer 1

Answer:

If E is the point of intersection of this bisector and the line through A and B,find the coordinates of E. Hence show that OA:OB = AE:EB. +1 vote.

Answer 2

Angle OAB's bisector's equation is y=x.

If E is where this bisector and the line between A and B connect, then E's

coordinates are (40/3,40/3)

We can say OA because E is the location where the bisector of the angle

OAB and the lines A and B connect.

OB= AE : EB

Since point O is the origin, we may state that its coordinates are (0, 0),

and the angle OAB's bisector produces an angle of 45 degrees with

respect to the x axis, according to the formula for a line passing through

the origin (y= mx).

where,

Angle produced by the bisector with respect to the x axis is equal to (45

degree)

m=tan θ

applying  substitution method

m=tan45

m=1

the equation of the bisector of angle OAB is  y=x

Now, the equation of the line passing through point A (5,0) and point B

(0,8) by formula,

(y - y1 ) = { ( y2 - y1 ) / ( x2 - x1 ) } *( x - x1 )

here , y1 = 0, x1 = 5, y2 = 8, x2 = 0

again, by substitution method,

( y - 0 ) = { ( 8 - 0 ) / ( 0 - 5 ) } * ( x - 5 )

Y = { 8 / -5 } *  ( x - 5 )

8x - 5y = 40

point E is the intersection of 8x - 5y = 40 and y = x the solving these two

equations by substitution method we get;

x = 40 / 3

y = 40 / 3

the co-ordinates of point E are ( 40 / 3 , 40 / 3 )

E is the intersection point of the bisector of the angle OAB and the line

through point A and B on comparing triangle OEA and triangle OEB we

can say that,

OA : OB= AE : EB

the angle OAB's bisector has the equation y = x.

Now, using the formula ( y - y1 ) = ( y2 - y1 ) / ( x2 - x1 ), the equation of the line

passing through points A (5, 0) and B (0, 8) can be found.

( y - y 1 ) = { ( y2 - y1 ) / ( x2 - x1 ) } * ( x - x 1 )

Y1 = 0, X1 = 5, Y2 = 8, X2 = 0 in this case.

Once more using the substitution method,

(y - 0) = ( 8 - 0) / ( 0 - 5 ) * ( x - 5 )  \ s

y= {8/-5} *(x-5)

40 points = 8x - 5y by using the substitution method to solve these two

equations, we obtain x = 40 / 3 and y = 40 / 3 as the intersection of

8x - 5y = 40 and y = x.

Point E's coordinates are ( 40/3 , 40/3 )

By comparing triangles OEA and OEB, we can say that OA : OB= AE : EB

E is the point at where the bisector of the angle OAB and the line

between points A and B cross.

Hence, OA:OB= AE:EB

Learn more about substitution method here  https://brainly.com/question/14619835

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