Transcript 5.4 Medians and Altitudes
5.4 Medians and Altitudes
Say what?????
Vocabulary…
Concurrent- 3 or more lines, rays, or segments that intersect at the same point Median of a Triangle – a segment from a vertex to the midpoint of the opposite side Centroid – point of concurrency of 3 medians Altitude of a Triangle – the perpendicular segment from a vertex to the opposite side or to the line that contains the opposite side – there are 3 in a
Orthocenter - the point where the 3 altitudes of a intersect
Theorem 5.8: Concurrency of Medians of a
-the medians of a intersect at a point that is 2/3 the distance from each vertex to the midpoint of the opposite side.
Theorem 5.9: Concurrency of Altitudes of a
-the lines containing the altitudes of a are concurrent
In
RST
,
Q
is the centroid and
SQ
= 8
. Find
QW SW
.
and
SQ
= 2 3
SW
8 = 2 3
SW
Concurrency of Medians of a Triangle Theorem Substitute
8
for
SQ
.
12 =
SW
Multiply each side by the reciprocal, .
2
Then
QW
=
SW – SQ =
12 – 8 = 4.
So,
QW
= 4
and
SW =
12
.
With a partner, do #’s 1-2 on p.278
With a partner, do #’s 3-4 on p.279
Practice
In Exercises
1–3,
use the diagram.
G
is the centroid of
ABC
.
1.
If
BG
= 9,
find
BF
.
ANSWER
13.5
2.
If
BD
= 12,
find
AD
.
ANSWER
12
3.
If
CD
= 27,
find
GC
.
ANSWER
18
Find the orthocenter.
Find the orthocenter
P
in an acute, a right, and an obtuse triangle. (Draw 3 altitudes…drop perpendicular lines from vertex to opposite side.) SOLUTION Acute triangle Right triangle Obtuse triangle
P
is inside triangle.
P
is on triangle.
P
is outside triangle.
Can you answer these??????
Look at the and answer the following: 1. Is BD a median of ABC?
2. Is BD an altitude ABC?
3. Is BD a perpendicular bisector?
B B A D C A D C