Download COORDINATE GEOMETRY Find the coordinates of the centroid of

5-2 Medians and Altitudes of Triangles
COORDINATE GEOMETRY Find the
coordinates of the centroid of each triangle with
the given vertices.
11. A(–1, 11), B(3, 1), C(7, 6)
12. X(5, 7), Y(9, –3), Z(13, 2)
SOLUTION: The midpoint D of
SOLUTION: The midpoint D of
is Note that is a line that is Note that is a line that connects the vertex C and D, the midpoint of
. The distance from D(1, 6) to C(7, 6) is 7 – 1 or 6
units.
If P is the centroid of the triangle ABC, then
So, the centroid is connects the vertex Z and D, the midpoint of . The distance from D(7, 2) to Z(13, 2) is 13 – 7 or 6
units.
If P is the centroid of the triangle XYZ, then
So, the centroid is or 4 units to the
left of Z. The coordinates of the centroid(P) are (13–
4, 2) or (9, 2).
or 4 units to the left of C. The coordinates of the centroid (P) are
(7 – 4, 6) or (3, 6).
COORDINATE GEOMETRY Find the
coordinates of the orthocenter of each triangle
with the given vertices.
14. J(3, –2), K(5, 6), L(9, –2)
SOLUTION: 12. X(5, 7), Y(9, –3), Z(13, 2)
The slope of
is or So, the slope of SOLUTION: The midpoint D of
the altitude, which is perpendicular to
is Note that is a line that connects the vertex Z and D, the midpoint of . The distance from D(7, 2) to Z(13, 2) is 13 – 7 or 6
units.
If P is the centroid of the triangle XYZ, then
So, the centroid is or 4 units to the
left of Z. The coordinates of the centroid(P) are (13–
4, 2) or (9, 2).
is Now, the equation of the altitude from L to
is:
Use the same method to find the equation of the
altitude from J to
.That is,
Solve the equations to find the intersection point of
the altitudes.
eSolutions Manual - Powered by Cognero
Page 1
5-2 Medians and Altitudes of Triangles
COORDINATE GEOMETRY Find the
coordinates of the orthocenter of each triangle
with the given vertices.
14. J(3, –2), K(5, 6), L(9, –2)
SOLUTION: The slope of
is or So, the slope of the altitude, which is perpendicular to
15. R(–4, 8), S(–1, 5), T(5, 5)
SOLUTION: The slope of
is or –1. So, the slope of
the altitude, which is perpendicular to
the equation of the altitude from T to
is 1. Now,
is:
is Now, the equation of the altitude from L to
Use the same way to find the equation of the altitude
is:
from R to
.That is,
Solve the equations to find the intersection point of
the altitudes.
Use the same method to find the equation of the
altitude from J to
.That is,
So, the coordinates of the orthocenter of
4, –4).
is (–
Solve the equations to find the intersection point of
the altitudes.
ALGEBRA Use the figure.
So, the coordinates of the orthocenter of
–1).
is (5,
24. If
m
is an altitude of
2 = 3x + 13, find m
, m 1 = 2x + 7,and
1 and m 2.
SOLUTION: By the definition of altitude,
Substitute 14 for x in
15. R(–4, 8), S(–1, 5), T(5, 5)
eSolutions Manual - Powered by Cognero
SOLUTION: Page 2
is not an altitude of 5-2 Medians and Altitudes of Triangles
because m
ECA =
92.
ALGEBRA Use the figure.
24. If
m
is an altitude of
2 = 3x + 13, find m
CCSS ARGUMENTS Use the given
information to determine whether
is a perpendicular bisector, median, and/or
an altitude of
.
, m 1 = 2x + 7,and
1 and m 2.
SOLUTION: By the definition of altitude,
27. SOLUTION: is an altitude by the definition
Since
,
of altitude.We don't know if it is a perpendicular
bisector because it is not evident that M is the
midpoint of
Substitute 14 for x in
28. SOLUTION: Since
, we know that
by CPCTC.
Since
and they are a linear pair,
then we know they are right angles and
.
Therefore,
is the perpendicular bisector, median,
and altitude of
.
25. Find the value of x if AC = 4x – 3, DC = 2x + 9, m
ECA = 15x + 2, and
.
is a median of Is
? Explain.
also an altitude of 29. SOLUTION: Since we know that
midpoint of
. Therefore,
. SOLUTION: Given: AC = DC.
30. then M is the
is the median of
and
SOLUTION: Since Substitute 6 for x in m
is not an altitude of and
, we can prove
by HL. Therefore, we know
that
by CPCTC, making M the midpoint
of
. Therefore,
is the perpendicular bisector,
median, and altitude of
.
ECA.
because m
ECA =
92.
CCSS ARGUMENTS Use the given
information
to determine
eSolutions
Manual - Powered
by Cognerowhether
is a perpendicular bisector, median, and/or
an altitude of
.
Page 3