Embed Size (px)
Citation preview
Centers of Triangles Packet
1
Name _______________________________
1) If Q is the vertex angle of isosceles PQR , and RA is a median, find QRm 2) Which side is the dot closest to? Can you draw a point that is the same distance away from the 3 sides? This point is called the incenter. Add markings to indicate that point P is the triangle’s incenter. But… Why is it called the INcenter???
Q
A
P R
4 inches
A
B
C
P
Centers of Triangles Packet
2
3) Sketch the incenter of each triangle and show markings in two different ways that indicate it is the incenter. 4) In the triangle below, point P is the incenter. Find the measures of angles x, y, and z. 5) In the triangle below, point Q is the incenter. Find the measures of angles w, x, y, and z.
30o
24o
x y
z
Q 18o
34o
w x
y
z
P
Centers of Triangles Packet
3
6) In the triangle below, point P is the incenter. Find the measures of angles x, y, and z. 7) In the triangle below, point Q is the incenter. Find the measures of angles w, x, y, and z. 8) In isosceles ABC , BCABAC . If P is the triangle’s incenter, find the measures of angles w, x, y, and z.
P 26o
22o
x y
z
Q 21o
23o
w
x
y
z
P
A
B
C
34o w x
y z
Centers of Triangles Packet
4
9) Which vertex is the dot closest to? 10) Can you draw a point that is the same distance away from the 3 vertices?
This point is called the circumcenter. 11) What can you add to your diagram to show that you drew the circumcenter?
Add markings to indicate that point P is the triangle’s circumcenter. But… Why is it called the CIRCUMcenter???
A
B
C
P
Centers of Triangles Packet
5
12) In each triangle below, add markings to show that the given point is the circumcenter, and construct the circumscribed circle. 13) If P is the circumcenter of the triangle below, which of the following choices must be correct? (Circle all that apply)
P
A
B
C
A]
B]
C]
D]
E] is isosceles.
F] If , then
G]
Centers of Triangles Packet
6
14) If Q is the circumcenter of the triangle below, which of the following choices must be correct? (Circle all that apply) 15) Choose incenter or circumcenter for each of the following:
Q
A
B
C
D
A]
B]
C]
D]
E] is isosceles.
F]
P
A
B
C
D E
F
a] If P is the center of the triangle’s inscribed circle, then point P is the (incenter/circumcenter) b] If P is the center of the triangle’s circumscribed circle, the n point P is the (incenter/circumcenter) c] If the 3 sides of the triangle are bisected, point P is the (incenter/circumcenter) d] If the 3 angles of the triangle are bisected, then point P is the (incenter/circumcenter)
e] If , then point P is the (incenter/circumcenter)
f] If , then point P is the (incenter/circumcenter)
Centers of Triangles Packet
7
16) Determine whether each point shows the triangle’s incenter, or circumcenter. a] b] c] d] e] f] 17) The centroid of a triangle is the point where the 3 medians intersect. 18) a] The incenter of a triangle is the point where the 3 ___________________________ intersect. It is equidistant from the triangle’s 3 _____________________________. It is the center of the triangle’s _______________________________________________. b] The circumcenter of a triangle is the point where the 3 _________________________ intersect. It is equidistant from the triangle’s 3 _____________________________. It is the center of the triangle’s _______________________________________________.
B A C
D
E
25o 25o
35o
35o
30o 30o
Centers of Triangles Packet
8
c] The centroid of a triangle is the point where the 3 _____________________________ intersect. 19) The centroid has a special feature… Can you figure it out? 20) 21)
Centers of Triangles Packet
9
22) In ABC , 9AFm , and 4.2GEm . Find AGm and CEm .
23) In the diagram, 172 xMSm , 264 xNSm , and 27SPm .
24) In LMN , 21RLm , and 4SQm . Find LSm and NQm
A
B
C D
E F
M
N
G
L
R P
S
Q
M
N
L
P S
Q
a] Find x. a] Find PS b] Find PM c] Find NS d] Find SQ
Centers of Triangles Packet
10
25) In GHJ , 9.2PAm , and 8.10HCm
26) In LMN , 7SPm , and 1.8NQm . Find MPm and NSm
27) In the diagram, 10CDm , and 6.12GBm
G
H
J
A B
P
C
M
N
L
R P
S
Q
a] Find PC b] Find HP c] Find JA d] Find JP
a] Find GD b] Find GC c] Find FG d] Find FB
Centers of Triangles Packet
11
28) Bowser: “The centroid is the point where the triangle’s 3 medians intersect.”
Mario: “The centroid divides each median into a ratio of 1:2” Luigi: “The centroid is located 2/3 of the way from each vertex to its opposite side. Who is correct? 29) Determine whether each picture shows the triangle’s incenter, circumcenter, or centroid. a] b] c] d] e] f]
A B
C
E
25o 25o
30o 30o
35o
35o
Centers of Triangles Packet
12
30) The orthocenter is the point where the triangle’s three ___________________________ intersect. 31) Name the 3 properties of the incenter, and provide diagrams to help you study. a] b] c]
Centers of Triangles Packet
13
32) Name the 3 properties of the circumcenter, and provide diagrams to help you study. a] b] c] 33) Name 2 properties of the centroid, and provide diagrams to help you study. a] b]
Centers of Triangles Packet
14
34) Name 1 property of the orthocenter, and provide a diagram to help you study. a] 35) Which of the following is NOT true about the incenter? (1) It is equidistant from the triangle’s 3 sides. (2) It is the point of concurrency of the triangle’s 3 angle bisectors. (3) It is the center of the triangle’s inscribed circle. (4) It is the center of the triangle’s circumscribed circle. 36) Which of the following is NOT true about the circumcenter? (1) It is the center of the triangle’s circumscribed circle. (2) It is the point of concurrency of the triangle’s 3 perpendicular bisectors. (3) It is equidistant from the triangle’s 3 vertices. (4) None of the above. 37) Which of the following is NOT true about the centroid? (1) It is the point of concurrency of the triangle’s 3 medians. (2) It divides each median into a ratio of 1:2 (3) It is located 2/3 of the way from each vertex to its opposite side. (4) It is located 2/3 of the way from each side to its opposite vertex.
Centers of Triangles Packet
15
38) What does the point G represent? State 3 facts about it. 39) Will the __________________________________ always be inside the triangle? 40) What does the point G represent? State 2 facts about it. 41) Will the __________________________________ always be inside the triangle?
A
B
C D
E F G
Centers of Triangles Packet
16
42) What does the point Z represent? State 3 facts about it. 43) Will the __________________________________ always be inside the triangle? 44) What does the point H represent? How did you know? 6) How do you know 45) Will the __________________________________ always be inside the triangle?
Z
Centers of Triangles Packet
17
46) Identify the center shown in each diagram, and explain how you knew. 47) Consider the diagram below:
a] What is the point G called?
b] It must be that (TRUE / FALSE)
c] It must be that (TRUE / FALSE)
d] It must be that (TRUE / FALSE)
e] If , and , what is
?
Centers of Triangles Packet
18
48) Consider the diagram below: 49) Consider the diagram below:
Q R
S
a] What is the point B called?
b] It must be that (TRUE / FALSE)
c] It must be that (TRUE / FALSE) d] B would be located outside of the triangle if the triangle was… (1) acute (2) obtuse e] If this were a right triangle, then B would be located: (1) On the hypotenuse of the triangle (2) On the vertex of the right angle of the triangle
a] What is the point S called?
b] If , find and .
c] If , find and .
d] It must be that (TRUE / FALSE)
e] It must be that (TRUE / FALSE)
