Download King High School February Invitational Team Round 1. Circle O has

King High School February Invitational
Team Round
1. Circle O has radius 2.
A= the area of O
B= the length of the longest chord in O
C= the perimeter of a 120o sector of O
D= the area of an inscribed equilateral triangle in O
Find ( A3 + B − C)D.
King High School February Invitational
1. Circle O has radius 2.
A= the area of O
B= the length of the longest chord in O
C= the perimeter of a 120o sector of O
D= the area of an inscribed equilateral triangle in O
Find ( A3 + B − C)D.
Team Round
King High School February Invitational
Team Round
2. With given information in the diagram, how many of the following statements must be true
regarding the figure?
I. 4ABC ∼ 4A0 B 0 C 0
IV. B 0 C + B 0 C 0 < CC 0
II.
V.
A0 B 0
AB
AB
PB
=
=
B0C
P B0
P C0
PC
III. AB k A0 B 0
VI. m6 BP C = m6 B 0 P C 0
King High School February Invitational
Team Round
2. With given information in the diagram, how many of the following statements must be true
regarding the figure?
I. 4ABC ∼ 4A0 B 0 C 0
IV. B 0 C + B 0 C 0 < CC 0
II.
V.
A0 B 0
AB
AB
PB
=
=
P C0
PC
B0C
P B0
III. AB k A0 B 0
VI. m6 BP C = m6 B 0 P C 0
King High School February Invitational
Team Round
3. The incircle of an equilateral triangle has area 3π.
A= the side length of the triangle
B= the area of the triangle
C= the perimeter of one of the three regions outside the circle but inside the triangle
What is A × B × ( C2 − 3)?
King High School February Invitational
Team Round
3. The incircle of an equilateral triangle has area 3π.
A= the side length of the triangle
B= the area of the triangle
C= the perimeter of one of the three regions outside the circle but inside the triangle
What is A × B × ( C2 − 3)?
King High School February Invitational
√
4. A regular octagon has side length 2 inches.
A= the number of sides in an octagon
B= the sum of the degree measures of the octagon’s interior angles
C= the area of the octagon in square inches
.
Find BC
A
Team Round
King High School February Invitational
√
4. A regular octagon has side length 2 inches.
A= the number of sides in an octagon
B= the sum of the degree measures of the octagon’s interior angles
C= the area of the octagon in square inches
Find BC
.
A
Team Round
King High School February Invitational
Team Round
5. Equilateral triangles 4AEB and 4DCF are constructed on sides AB and CD, respectively,
of trapezoid ABCD. AC and BD intersect at G. Add the values of the true statements and
subtract the values of the false statements.
(2) E must be on line BC.
(8) m6 ACF = m6 CAE
King High School February Invitational
(4) AG(GC) = BG(GD)
(16) G must be on the line EF .
Team Round
5. Equilateral triangles 4AEB and 4DCF are constructed on sides AB and CD, respectively,
of trapezoid ABCD. AC and BD intersect at G. Add the values of the true statements and
subtract the values of the false statements.
(2) E must be on line BC.
(8) m6 ACF = m6 CAE
(4) AG(GC) = BG(GD)
(16) G must be on the line EF .
King High School February Invitational
Team Round
6. 4ABE is a 30-60-90 right triangle. GD is a midline of 4ABE, and HC is a midline of
trapezoid AGDB. 4F GE is equilateral.
A=the value of
FG
AG
C=the perimeter of 4AF E if AB = 1
B=the measure of 6 AF E in degrees
D=the area of trapezoid HGDC if AB = 1
B
Find A( 90
+ 2C − 32D).
King High School February Invitational
Team Round
6. 4ABE is a 30-60-90 right triangle. GD is a midline of 4ABE, and HC is a midline of
trapezoid AGDB. 4F GE is equilateral.
A=the value of
FG
AG
C=the perimeter of 4AF E if AB = 1
B
+ 2C − 32D).
Find A( 90
B=the measure of 6 AF E in degrees
D=the area of trapezoid HGDC if AB = 1
King High School February Invitational
Team Round
7. A triangle has sides a = 13, b = 14, and c = 15.
A= the perimeter of the triangle
B=the height of the triangle to side b
C=cosA
D=the area of the triangle
Find B
+D
C
A
King High School February Invitational
7. A triangle has sides a = 13, b = 14, and c = 15.
A= the perimeter of the triangle
B=the height of the triangle to side b
C=cosA
D=the area of the triangle
Find B
+D
C
A
Team Round
King High School February Invitational
Team Round
AE
8. In 4ABC, cevians AF , CD, and BE intersect at P such that EC
= 3 and F is the midpoint
of BC. A cevian is any line segment in a triangle with one endpoint on a vertex of the triangle
and the other endpoint on the opposite side. [4ABC] refers to the area of 4ABC.
[4ABE]
A= [4BEC]
Find
]
B= [4ABP
[4AP C]
AD
C= DB
D= PAPF
(A+C)B
.
D
King High School February Invitational
Team Round
AE
8. In 4ABC, cevians AF , CD, and BE intersect at P such that EC
= 3 and F is the midpoint
of BC. A cevian is any line segment in a triangle with one endpoint on a vertex of the triangle
and the other endpoint on the opposite side. [4ABC] refers to the area of 4ABC.
[4ABE]
A= [4BEC]
Find
(A+C)B
.
D
]
B= [4ABP
[4AP C]
AD
C= DB
D= PAPF
King High School February Invitational
Team Round
9. Circle ω1 and circle ω2 intersect at points A and B. Points E, C on ω1 and points D, F on ω2
are all on EF , which intersects AB at P . How many of the following statements are always
true regarding the figure?
I. F C(F E) = F A2
II. CP (P E) = DP (P F )
III. AP (P B) = DP (P C)
IV. ED(EF ) = EB(EA)
King High School February Invitational
Team Round
9. Circle ω1 and circle ω2 intersect at points A and B. Points E, C on ω1 and points D, F on ω2
are all on EF , which intersects AB at P . How many of the following statements are always
true regarding the figure?
I. F C(F E) = F A2
II. CP (P E) = DP (P F )
III. AP (P B) = DP (P C)
IV. ED(EF ) = EB(EA)
King High School February Invitational
Team Round
10. In 4ABC, a = 6, b = 8, and 6 C = 30o .
A= the number of possible lengths of side c
B= m6 A + m6 B in degrees
C= the area of 4ABC in square units
D= the value of c2
Find A2 + B + C + D.
King High School February Invitational
10. In 4ABC, a = 6, b = 8, and 6 C = 30o .
A= the number of possible lengths of side c
B= m6 A + m6 B in degrees
C= the area of 4ABC in square units
D= the value of c2
Find A2 + B + C + D.
Team Round
King High School February Invitational
Team Round
11. Points A, B, C and D lie on a circle. AC intersects BD at point P , and AB intersects CD at
d have measure α and the inscribed
point Q. Let the inscribed angles subtending minor arc BC
d have measure β. Answer in terms of α and β.
angles subtending minor arc AD
d
A= the measure of the central angle subtending minor arc AD
B= the measure of 6 BP C
C= the measure of 6 AQC
D= the measure of 6 ADB if AB = DC
Find A + B + C + D.
King High School February Invitational
Team Round
11. Points A, B, C and D lie on a circle. AC intersects BD at point P , and AB intersects CD at
d have measure α and the inscribed
point Q. Let the inscribed angles subtending minor arc BC
d have measure β. Answer in terms of α and β.
angles subtending minor arc AD
d
A= the measure of the central angle subtending minor arc AD
B= the measure of 6 BP C
C= the measure of 6 AQC
D= the measure of 6 ADB if AB = DC
Find A + B + C + D.
King High School February Invitational
Team Round
12. 4ABC has integral side lengths and a right angle at B. 4DEF is formed by the midpoints
.
of the sides of 4ABC and has area 15
2
A= the area of 4ABC
B= the perimeter of 4DEF
C= the perimeter of 4ABC
D= the area of the incircle of 4ABC
D
)B.
Find ( A+C
King High School February Invitational
Team Round
12. 4ABC has integral side lengths and a right angle at B. 4DEF is formed by the midpoints
of the sides of 4ABC and has area 15
.
2
A= the area of 4ABC
B= the perimeter of 4DEF
C= the perimeter of 4ABC
D= the area of the incircle of 4ABC
D
Find ( A+C
)B.
King High School February Invitational
Team Round
13. A= the number of sides in a dodecagon
B= the number of diagonals in a decagon
C= the sum of the interior angles of a nonagon in degrees
D= the area of a hexagon with side length 1
.
Find the value of CD
AB
King High School February Invitational
13. A= the number of sides in a dodecagon
B= the number of diagonals in a decagon
C= the sum of the interior angles of a nonagon in degrees
D= the area of a hexagon with side length 1
Find the value of CD
.
AB
Team Round