Example 2
Perpendicular Lines
Write an equation of the line perpendicular to 3x + y - 6 = 0 and
passing through the point P(5, 2).
Solution
To use the point-slope form, you need a point on the line and the slope of
the line. The slope of the new line is the negative reciprocal of the slope of
3x + y - 6 = 0, because the lines are perpendicular.
To find the slope of 3x +y - 6 = 0, write the equation in the form y = mx + b.
3x+ y- 6 = 0
y =-3x + 6
The slope·of 3x + y- 6 = 0 is -3.
The slope, m, of a line perpendicular to 3x + y - 6 = 0
is the negative reciprocal of -3, which is~·
A point on the line is P(5, 2).
· Use the point-slope form:
y-y1 =m(x-x)
Substitute known values:
y- 2 = - (x- 5)
Expand:
Multiply both sides by 3:
Write in standard-form:
1
-
3
1
5
y- 2 =-x- 3
3
3y - 6 =x - 5
0 = X- 3y + 1
An equation of the line perpendicular to 3x + y - 6 = 0 and passing ~r_ough
the point P(5, 2) is x- 3y + 1 = 0.
-· ~actice
W ·Determine an equation for each of the following lines.
the line parallel toy = 3x + 4 and passing through the point (2, 1)
--b}' the line parallel to 2x - y = 7 and passing through the point (- 3, 2) ~::. • --~ 4 ~
c) the line parallel to x +- 2y - 5 = 0 and passing through the point (5, - 3)
--(d)' the line parallel to 3:- 9y- ~ = 0 ~nd having the same y-interc~pt ~ \
as 2x + y - 8 = 0
I.A.., ·\: - > ;< .._ 1
t
/
:: t-1 r- c
--a)
f~, Determine an equation fur each of the followi~g lines.
~ the line perpendicular toy= -2x + 4 and passing through the
\ _
'
~.
point (4, 6)
._ b), the line perpendicular to x- 3y - 1 = 0 and passing through the • _,
7
point (1, - 1)
C) the line perpendicular tOX+ 2y- 8 = 0 and passing through the
point (- 3, -4)
...._:.d~ the line perpendicular to 4x- 2y + 3 =0 and having the same
I ./ •
_ x-mtercept as 2x + 3y- 10 = 0
_
·
-.'
•.•
· ':J
or ""
'
~
: _ )(
~
f
:
• 0--- ',
Review: Equations ofLines 87
/:)Determine an equation for the right bisector of the line segment joining
~3, 6) and B(-1, 2).
~erify that the given point lies on the perpe~dicular bisector of the given
~ .s.egment.
.
.
--a) point 'A(3, 4); line segment BC, with _
endpoints B(2, 1) and C(6, 5)
b) point P(1, 3); line segment QR, with endpoints Q(-3, 1) and R(3 , -1)
c) point K(-2, -4); line segment LM, with endpoints L(O, 2) and M(4, -6)
l = 10. The points
C(3, 1) and D(1, -3) are the endpoints of chord CD. EF right bisects chord
CD at G. Verify that the centre of the circle lies on the right bisector of
chord CD.
,. ,·
"'.
8. The equation of a circle with centre 0(0, 0) is i +
';<
\)
>- ' (
9. The vertices of a quadrilateral are A(O, 0), B(2,
r
'
r
Verify that the diagonals of ABCD are perpendicular to each othe~.
•'1
'I
h C(5, 1), and D(3, -2).
,/
)( '•
\.
I.
10. V~f¥ that the quadrilateral with vertices _Q(D,-O)rE(3 ,_~), Q(13., 7),
/
and R(), f) is a trapezoid.
__ ..
~-
----
__.
~----
. · -,
·,1
'
X'/ ~
7.
( -n~ifyl:h<Irthf--q~hdrilateral with vertices P(-2, 2), Q(- 2, -3),
R(-5; -hand S(-5; 0) is a parall~lograrn
-
___
. ......
----- -- ~- -- .------12. A triangle has vertices K(-2, 2), L(1, 5), and M(3, -3). Verify that
a)
the triangle has a right angle
b) the midpoint of the hypotenuse is the same distance from each vertex
13. Quadrilateral PQRS has vertices P(O, 6), Q(- 6, -2), R(2, - 4), and
S(4, 2). Verify that the quadrilateral formed by joining the midpoints of
the sides of PQRS is a parallelogram.
~ .6ABC has vertices A(3, 4), B(- 5, 2), and C(1, -4). Determine an
~ation for
a)
CD, the median from C to AB
b) AE, the altitude from A to BC
-f
GH, the right bisector of AC
15. A triangle has vertices X(O, 0), Y(4, 4), and Z(8, -4).
a)
Write an equation for each qf the three medians.
b) Recall that the centroid of a triangle is the point of intersection of
the medians of the triangle. Use the equations from part a) to verify that
(4, 0) is the centroid of .6XYZ.
16. .6AOB has vertices A(4, 4), 0(0, 0), and B(8, 0). EF right bisects
AB at P. GH right bisects OA at Q . Determine the coordinates of the
circumcentre of .6~ ·
l\C~
17 . .6POR has vertices P(O, 6), 0(0, 0), and R(6, 0). Determine the
coordinates of the centroid of .6POR.
96 Chapter 2