Preparing for the Regents Examination Geometry, AK

8-6 Concurrence of the
Altitudes of a Triangle
(the Orthocenter)
(page 168)
1 (4) obtuse
2 (4) at one of the vertices of the triangle
Exercises 3–9: Check students’ graphs.
3 (2.2, 1.2)
5
4 ( _ , 4) 3
5 (1, 0)
6 (6, 6.5)
7 ( 2,
22 _
3 )
8 (1, 1)
9 (1, 0.5)
10 a Slope of −−
AB is 4 _ . Slope of
3 −−
BC is 3 _ . Since
4
the slopes are negative reciprocals, the
segments are perpendicular and B is the
right angle.
b −−
AB and −−
BC can each be altitudes. B is the
endpoint of the altitude drawn from B
to AC.
Chapter Review (pages 168–171)
1 (2) y 3 _ x 4
4
2 (2) (0, 3)
3 (4) (1, 9)
4 (2) y 5x 1
5 (1) y 3x 3
6 No. Sub-in: 3(5) 4(2) 0
7 x-intercept is (2, 0) and y-intercept is (0, 3).
Distance is √ 13 .
8 x-intercept is (15, 0) and y-intercept is (0, 3).
Distance is 15.3.
9 a 10
10 B(10, 10)
11 Slope of ___
OB is 1 _ . Slope of
2 ___
AC is 2. Their
product is ( 1 _
2 ) (2) 1.
12 a (0, 3) b (4, 0) c (8, 3)
13 a m 3 b x-intercept is (4, 0)
c y-intercept is (0, 12)
d (3, 3) is not on the line.
14 a neither b vertical c vertical
d neither e horizontal f vertical
15 a 1 _ b 2 c
2 3 _
4
d 2 _ e
3
2 _ f
3
a _
d
16 a no slope b 0 c 1
d 7 _ e
d b
2 _
c a
f
k _
h
17 a 7 b 2 c 14 d 2
e 6 f 5 g 4
18 a not collinear
b collinear
c collinear
d collinear
e collinear
f collinear
19 a y 3
b y 3 _ x
2 9 _
2
c y 1 _ x
3 5 _
3
d y 2 _ x
3
e y 3 _ x
2 19 _
2
20 a AB AC 2 √ 10
b Midpoint of ___
BC is (2, 1). Slope of the
median from A to ___
BC slope of altitude
from A to ___
BC 1.
21 a (3, 2) b (2, 3) c (6.5, 8.5)
22 Slope of ___
DR slope of ____
AW 1 _
. Slope of
3 ___
RA is not equal to the slope of ____
DW .
___
RA ____
DW 5
23 a NO 5 √ 2 and OP 2 √ 5
b Slope of ___
EO is 1 _ . Slope of
3 ___
PN is 3.
Slopes of perpendicular lines are negative
reciprocals of each other.
24 A(0, 3), B(4, 2), C(5, 4), and D(1, 3). ABCD
is a parallelogram because both pairs of opposite
sides are parallel. Slope of ___
AB slope
of ___
CD 1 _ . Slope of
4 ___
BC slope of ___
DA 6.
25 QRST is a parallelogram because both pairs
of opposite sides are parallel. Slope of
___
QR slope of ___
ST b _
a
. Slope of ___
RS slope
of ___
TQ 0.
26 a–b Check students’ graphs.
c (2, 10)
Chapter Review 43