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# Equations of Lines and Intersections

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The diagram below (see attachment) shows a triangle ABC whose vertices are at A (-1, 3), B (6, 5) and C (8, -3). The line BP is perpendicular to the line AC, and M is the midpoint of BC.

Note that BP is called an altitude of triangle ABC and that AM is called a median of triangle ABC.

i) The line AC
ii) The altitude BP
(Show answers as a fraction in its simplest form)

b) Find the equation of the altitude BP, expressing your answer in the form: y = mx + c

c) Find the coordinates of M.

d) Find the equation of the median AM, expressing your answer in the form: y = mx + c

e) Explain how you would find the coordinates of the point of intersection of the altitude BP and the median AM.

https://brainmass.com/math/triangles/equations-lines-intersections-35782

## SOLUTION This solution is FREE courtesy of BrainMass!

gradient of a line (also called slope ) = (y2-y1) /(x2-x1)

i) The line AC
Gradient of a line (also called slope ) = (y2-y1) /(x2-x1)

A (-1,3) , C (8,-3)

Therefore gradient ={ (-3)-(3) }/{ (8)-(-1)} = -6/9 = -2/3

Answer: slope of line AC= -2/3
ii) The altitude BP

Altitude BP is perpendicular to AC
The product of the gradients of perpendicular lines = -1
Therefore ,
Slope of line AC x slope of line BP = -1
Or ( -2/3) x slope of line BP = -1
Or slope of line BP = -1 / (-2/3) = 3 /2

Answer: slope of line BP= 3/2

(Show answers as a fraction in its simplest form)

b) Find the equation of the altitude BP, expressing your answer in the form
y = mx + c

y = mx + c
But we have calculated m = 3/2 in part a above
Thus
y = 3/2 x + c
This line passes through B whose coordinates are (6,5)
Thus the coordinates of B must satisfy the equation of line BP

Therefore
5 = 3/2 (6 ) + c
or 5= 9 + c
or c= -4

Therefore the equation of line BP is y= 3/2 x - 4

Answer: Equation of altitude BP is y= 3/2 x - 4

c) Find the coordinates of M.

The coordinates of a point which divides a straight line internally in the ratio m1:m2 is given by
x coordinate = (m1x1 + m2x2 ) / (m1+ m2)
y coordinate = (m1y1 + m2y2 ) / (m1+ m2)
Here m1=m2=1 (M is the midpoint of BC. )
x1= 6 , y1=5
x2= 8 , y2=-3

Therefore coordinates of M are
x coordinate = (6 + 8 ) / (2)= 7
y coordinate = {5+ ( -3 )} / (2)= 1

Thus the coordinates of M is (7,1)

d) Find the equation of the median AM, expressing your answer in the form
y = mx + c

The coordinates of
A( -1,3)
M (7,1)
Therefore slope= (1-3) / { ( 7 ) - (-1) } = -2 / 8 = - Â¼

Thus the equation of the line AM is y = - Â¼ x + c
This line passes through A( -1,3)
Therefore
3 = -(1/4 ) x (-1) + c
or c = 3 - Â¼ = 11/4

or Equation of line AM is
y= - Â¼ x + 11/4

Answer: Equation of line AM is y= - Â¼ x + 11/4

e) Explain how you would find the coordinates of the point of intersection of the altitude BP and the median AM.

We have two equations
Equation of line BP is y= 3/2 x - 4
Equation of line AM is y= - Â¼ x + 11/4

We will solve these equations simultaneously to find the coordinates of the point of intersection of the altitude BP and the median AM.

If we do so we get
3/2 x - 4 = - Â¼ x + 11/4
or 7 / 4 x = 27 /4
or x= 27/7

Substituting the value of x in y= 3/2 x - 4
y= 25/14
Therefore the coordinates of the point of intersection of the altitude BP and the median AM are ( 27/7 , 25/14)

This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!