___________
___________
Name
2.1
Q.
Length of a Line Segment
MATHPOWERTM 10, Ontario Edition,
pp.
• To find the length of a line segment joining
2 —xi)
l=J(x
2 +(y2 —y’)
.
2
• An
equation of
the circle with
centre
66—73
, y)
1
(x
,
2
and (x
0(0, 0) and radius r
1. Determine the length of the line segment
joining each pair of points. Express each length
as an exact solution and as an approximate
solution, to the nearest tenth.
a) (3, 7) and (—1, —5)
y2), use
is x
2
+
the formula
2 =
y
.
2
r
5. Communication Explain why APOR is a
right triangle.
b) (0, 5) and (6, 0)
2. Determine the radius of the circle with centre
(—5, 6) and point (2, —7) on its circumference.
Round the radius to the nearest tenth, if
Applications
6. The vertices of a right triangle are (2, 2), (5, 8),
and (—2, 4). Find the area of the triangle.
necessary.
3. Classify each triangle as equilateral, isosceles,
or scalene. Then, find each perimeter, to the
nearest tenth.
a) W(2, 3), X(—1, —2), Y(5, —2)
7. Three points, A(—3, 1), B(2, 4), and C(7, 7), lie
on a straight line. Show that B is the midpoint of
AC.
8. The coordinates of the endpoints of the
diameter of a circle are (—3, —5) and (3, 3). Find
the length of the radius of the circle.
1) A(—1, j i), B(—4, —1), C(2, —1)
—
9. a) Verify that the quadrilateral with vertices
D(2, 6), E(3, 3), F(—3, 1), and G(—4, 4) is a
rectangle.
4. Find the perimeter of parallelogram ABCD.
b) Determine the length of its diagonals, to the
tenth.
nearest
t.
Copyright © 2001 McGraw-Hill Ryerson Limited
Chapter 2
21
Name
Midpoint of a Line Segment
2.3
MATHPOWERTM 10, Ontario Edition,
pp.
75—80
To find the midpoint, M, of a line segment joining (xi,
2
(xl+x
y)
and (x
, y2), use the midpoint formula,
2
yl+y2
2
1. Determine the midpoint of each line segment
with the given endpoints.
a) (—6, 2) and (4, 8)
Applications
4. The endpoints of the diameter of a circle are
(—3, 11) and (2, 9). What are the coordinates of
the centre of the circle?
b) (1.5, 3) and (—6, —2.5)
c) (—200, —100) and (350, 600)
d) (‘
2
.Land(_..
2’ 4
4)
.
5. The endpoints of line segment MN are
M(—6, —10) and N(2, —2). Find the coordinates of
the point P on the line segment MN such that
MP:PN = 3:1.
e) (3a, 2b) and (—3a, 5b)
f) (—6a, 5b) and (ha, 0)
2. Find the midpoints of the sides of ADEF.
6. A square has vertices K(—4, 3), L(3, 4),
M(4, —3), and N(—3, -4).
a) Find the coordinates of the midpoint of each
side.
::E;:;x
b) Find the coordinates of the point of
intersection of the diagonals.
3. Communication One endpoint of a line
segment is D(5, —7). The midpoint of the line
segment is M(3.5, 1.5). Explain how to find the
coordinates of the other endpoint, E, of the line
segment.
c) Find the perimeter of the square formed by
joining the midpoints of the sides of square
KLMN.
7. Vertex V of AUVW has coordinates (4, 6). The
coordinates of the midpoint of UV are (1, 6), and
the coordinates of the midpoint of VW are (3, 2).
Find the coordinates of points U and W.
I
Copyright © 2001 McGraw-Hill Ryerson Limited
Chapter 2 23
V1
Practice
1. Determine an equation for each
of the following lines.
a) 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 1
)Oint (—3, 2)
c) the line parallel to x + 2
y 5 = 0 and passing through the point
(5, —3)
d) the line parallel to 3x
y 1 = 0 and having the sainey—intercep
9
t
as 2x +y —8 = 0
—
—
—
1•
A
2. Determine an equation for
each of the following lines.
a) the line perpendicular toy = —2x
+ 4 and passing through
the
pOint (4, 6)
b) the line perpendicular to x
y 1 0 and passing through the
3
point (1, —1)
c) the line perpendicular to x +
y 8 0 and passing through the
2
point (—3, —4)
d) the line perpendicular to 4x
+ 3
0 and having the same
x-interccpt as 2x +
10 = 0
—
—
—
—
y
3
—
Review: Equiztions ofLines 87
Practice
y_interCePt of each line.
i. Find the slope and
x—0
c) 5
y+
b) 4x+y6°
a)y3X+
4
f)
e)3X+6Y_8
0
x—2y
d) 8
iox+0.5y—
°
I) 3
Ii) 2x+3y°
5x—2y
°
g) 6
of the line that passes through the
2. Find the slope and y_intercePt
following pairs of pointS.
c) (—3, -1) and (1, 7)
b) (1, 2) and (-2, 5)
(2,4) and (4, 6)
f) (2, 0) and (4, —1)
e) (—2, —2) and (0, —1)
(4, -2) and (5, -5)
,4)and(12)
7
I) (
Ii) (3, 2) and (-2, 3)
g) (—2, —1) and (—3, —5)
4
o=
x—4y
13
Practice
standard form for the line that passeS through t
slope.
given point and has the
c) (—6.:3
b) (4,2);V15
(3,
2
5);
m
a)
1
f) (l.
e) (2, —8); vi =
d) (—2,l)1h14
3
1
1) (—1. -; . = -1.5
h) (—-i, 3);m
g) (4, —); vi = 1
i. Write
an equation in
given
:“:
—
—
standard form for the line through the given p::E
b) C—2, 9) and D(1, 0)
a) A(4, 5) and B(3, 7)
d) G(—2, —3) and H(3, —1)
c) E(—3, —6) and F(-5, —7)
f) L(0, 4) and M(-3, 0)
e) J(-2, 3) and K(2, 8)
line and the
3. Vrite an equation in standard form for the horizontal
line through each point.
(i
8
d)
c) (—5, —6)
b) (—3, 2)
a) (4, 5)
2.
\Vrite an equation in
—,
e)
f) (0, —9)
g) (—1, 0)
Ii)
(0, 0)
Ii{
Name
2.4
s
Verifying Properties of Geometric Figure
WER 10, Ontario Edition, pp. 88—99
M
THPO
MAT
verify
mine characteristics of geometric figures and to
The following formulas can beused to deter
geometric properties.
Slope of a line segment:
m
=
—
—
Length of a line
Midpoint
of a
1 = j(x
segment:
line
segment:
(xi
+ X2
—
x )2
1
Ya
+ (y2
—
yi
)2
+
2
y y
line:
a
of
tion
equa
the
of
Point-slope form
line:
a
of
tion
equa
Slope and y-intercept form of the
—
B,
1. Communication For any three points A,
oints
midp
and C, not in a line, M and N are the
prove
of AB and AC, respectively. How can you
thatMN
II BC and
MN=-BC?
2
=
m(x x)
y = mx + b
—
Applications
4. AABC has vertices A(3, 5), B(2, 3), and
C(5, 2).
a) Find the equations of the three altitudes.
of the
b) Find the intersection point of any two
altitudes.
c) Verify that this point (the orthocentre of the
triangle) is on the altitude not used in part b).
Ii
2. ADEF has vertices D(—1, 3), E(7, 1), and
F(4, 6). Classify the triangle as
a) isosceles or scalene
b) right-angled or not
3. The vertices of a quadrilateral are S(1, 2),
T(3, 5), U(6, 7), and V(4, 4). Verify each of the
following.
a) ST(JV is a parallelogram.
b) The diagonals of STLTV bisect each other,
c) STUV is a rhombus.
d) The diagonals of STUV are perpendicular to
each other.
24
Chapter 2
5. The sides of a triangle have the equations
2x—3y+13=O,3x+2y=O,andx+5y—26=O.
Verify that the triangle is an isosceles right
triangle.
6. A quadrilateral has vertices P(—3, 1), Q(3, 7),
R(9, 3), and S(—1, —1).
no
a) Verify that PQRS has no equal sides and
parallel sides.
QR,
b) Find the midpoints A, B, C, and D of PQ,
RS, and SF.
c) Verify that ABCD is a parallelogram.
Copyright © 2001 McGraw—Hill Ryerson Limited
Name
C)
•
2.5
Distance From a Point to a Line
MATHPOWERTM 10, Ontario Edition
, pp. 100-405
• To determine the distance from a given point to a line whose equatio
n is given,
a) write an equation for the perpendicular from the given point to the
given line
b) find the coordinates of the point of intersection of the perpendicular
and the given line
c) use the distance formula
1. Communication Explain how to find the
shortest distance from the point P(—5, 7) to the
line x = 3.
4. Find the shortest distance from the given
point to the given line. Round to the nearest
tenth, if necessary.
a) (0, 0) and y——x+5
I
2. In each case, write an equation for the line
that is perpendicular to the line with the given
equation, and passes through the given point.
b) (0, 0) and 15x
a) y =
c) (-4, —5) and y
-
x —7; (2,5)
—
=
—29
=
x —4
b) y——x—2; (—1,—?)
d) (6, 5) and 7x+y+23=0
c) 4x
Applications
—
—
7
=
0; (—5, 2)
0
5. A line has a y-intercept of 3 and an x-intercept
of 4. What is the shortest distance from the
origin to this line?
d) 2x+5y+3=0;(3,—4)
3. Find the exact value of the shortest distance
from the given point to the given line.
a) (0,0)andy=2x—10
-
6. a) Find the exact distance from the point
A(5, 7) to the line joining B(—2, 1) and C(4, —3).
b) Find the exact length of BC.
b) (0,O)and5x+12y—39=0
c) (3, —2) and y =
d) (5, —2) and 4x
[
—
+
1
x +9
3
—
+
14
c) Use your answers to parts a) and b) to fmd
the area of LABC.
=
0
Copight © 2001 McGraw-Hill Ryerson Limited
Cpter 2
25
Answers
CHAPTER 2
Analytic Geometry
2.1
Length of a Line Segment
2.3 Midpoint of a Line Segment
1. a)
‘Jiö;
1. a) (—1, 5)
12.6
b)
Ji;
7.8
2.14.8
d)
3. a) isosceles; 17.7
c) (75, 250)
e) (0, 3.5b)
f)
b) equilateral; 18
(5
2
—
5”
2)
2. (-5, 0), (1.5, -1.5), (-1.5, -5.5)
4. 26
5. length of PQ
length of QR
=
=
.I,
length
of PR=J; PQ
2 +QR
2 =18÷32=50=
2
PR
Thus, by the Pythagorean Theorem, APQR is a right
triangle,
3. Let the coordinates of E be (x
,
1
Substitute the
values into the midpoint formula,
(3.5, 1.5)—
and
yi
(xi
+
2
m+
5
‘
Then, solve for x
1
2
1
using these equations: x
+
5
=
2
6. 15 square units
yi
7. AB=-;BC’=J
8. 5
and
+(—7)
=1.5. (xi, yi)=(2,1O)
2
4. (—0.5, 10)
9. a) DE
2
DE
—
b) (—2.25, 0.25)
=
2
DC
+
GF
=
Thus, ZEDG
=
10
=
+
40
=
FE
=
GD
5 P(0, 4)
50
=
2
GE
6. a) (—0.5, 3.5), (3.5, 0.5), (0.5, —3.5), (—3.5, —0.5)
b) (0, 0)
c) 20
90°.
7. U(—2, 6), W(2, —2)
b) 7.1
2.4 Verifying Properties of Geometric
Figures
5. vertices: A(-2, 3), B(1, 5), C(-4, 6); AS
slope of AB =
slope of AC
AC
= ‘Ji’;
=
1. Find the coordinates of M and N using the
midpoint formula. Find the slope of MN and of BC. If
6. a) PQ”-J, slope=1; QR=JL slope=—;
the slopes are equal, the line segments are parallel.
RS = n/iT, slope
Find the length of MN and of BC
=;
SP
=
slope
=
I
using
the length
b) A(O, 4), B(6, 5), C(4, 1), D(—2, 0)
formula. The length of MN should be half the length
c) sLopesofAsand .DC
3
,slopesofAD
and
=
ofBC.
BC
=2
2. a) isosceles
2.5 Distance From a Point to a Line
3. a) slopes of ST and VU
1. The shortest distance is along the line through P
that is perpendicular to the vertical line x
;
2
slopes of SV and
=
3
= 3. The
length of the line segment joining P(-5, 7) and Q(3, 7)
b) right
b) The midpoint of both SU and TV is
is 8.
(, 2)
b) y=x—or2x-3y-I9=0
c) ST=’TU=UV=VS=
d) slope of TV =—1; slope of SU
37
c) y=——x——or3x’i’4y+7=0
4
4
5
d)
4. a) from A to BC: y = 3x —4; from B to AC:
2
5
1
9
y_—x+—;fr
omCtoAB:y=——x+—
3
3
2
2
2. a> y=—2x+9or2x+y.-9=0
b)
3. a)
‘I
4. a) 4.2
4
b) 3
c)
b) 1.7
c) 2.9
b)4
c)32
d)
d) 99
c)
1
(,
23”
‘77)
(17
23”
--J
satisfies the equation of the other altitude
5
6.a)
32
Chapter 2 27