NAME
5-2
DATE
PERIOD
Study Guide and Intervention
Medians and Altitudes of Triangles
Medians A median is a line segment that connects a vertex of a triangle to the midpoint
of the opposite side. The three medians of a triangle intersect at the centroid of the
triangle. The centroid is located two thirds of the distance from a vertex to the midpoint of
the side opposite the vertex on a median.
Example
In ABC, U is the centroid and
"
BU = 16. Find UK and BK.
2
BK
BU = −
3
2
16 = −
BK
3
3
,
6
24 = BK
#
$
4
Lesson 5-2
BU + UK = BK
16 + UK = 24
UK = 8
Exercises
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
In ABC, AU = 16, BU = 12, and CF = 18. Find
each measure.
1. UD
$
2. EU
&
%
#
12
6
16
'
3. CU
4. AD
5. UF
6. BE
"
&
In CDE, U is the centroid, UK = 12, EM = 21,
and UD = 9. Find each measure.
7. CU
+
6
8. MU
.
10. JU
11. EU
12. JD
Chapter 5
11
,
9
$
9. CK
12
%
Glencoe Geometry
NAME
DATE
5-2
Study Guide and Intervention
PERIOD
(continued)
Medians and Altitudes of Triangles
Altitudes
An altitude of a triangle is a segment from a vertex to the line containing the
opposite side meeting at a right angle. Every triangle has three altitudes which meet at a
point called the orthocenter.
y
# (7, 7)
Example
The vertices of ABC are A(1, 3),
B(7, 7) and C(9, 3). Find the coordinates of the
orthocenter of ABC.
Find the point where two of the three altitudes intersect.
"
(9, 3) $
(1, 3)
x
0
Find the equation of the altitude from
−−−
A to BC.
−−−
If BC has a slope of −2, then the altitude
1
has a slope of −
.
2
−−
C to AB.
−−
2
, then the altitude has a
If AB has a slope of −
3
3
slope of - −.
2
y - y1 = m(x – x1)
y - y1 = m(x - x1)
Point-slope form
1
y-3=−
(x – 1)
1
m=−
, (x1, y1) = A(1, 3)
2
Distributive Property
Simplify.
3
y - 3 = -−
(x - 9)
2
3
27
y - 3 = -−
x+−
2
2
33
3
y = - −x + −
2
2
3
m = -−
, (x1, y1) = C(9, 3)
2
Distributive Property
Simplify.
Solve the system of equations and find where the altitudes meet.
5
1
y=−
x+−
2
33
3
y = -−
x+−
2
2
2
5
33
3
1
−
x+−
=-−
x+−
2
2
2
5
33
− = −2x + −
2
2
2
2
2
2
2
2
33
Subtract −
from each side.
2
−14 = −2x
7=x
5
5
7
5
1
1
y = − x + − = − (7) + −
=−
+−
=6
2
1
Subtract −
x from each side.
Divide both sides by -2.
2
The coordinates of the orthocenter of ABC is (6, 7).
Exercises
COORDINATE GEOMETRY Find the coordinates of the orthocenter of each triangle.
1. J(1, 0), H(6, 0), I(3, 6)
Chapter 5
2. S(1, 0), T(4, 7), U(8, −3)
12
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2
1
1
y-3=−
x–−
2
2
5
1
y = −x + −
2
2
Point-slope form