2024年3月30日发(作者：)

0公式与定理：

垂直的等价公式 正弦定理 余弦定理 塞瓦定理 梅涅劳斯定理 中线长定理 角平分

线长定理 托勒密定理 海伦公式 圆内接四边形面积公式 费马点 三角形的五心

直线型

¯¯

1. In triangle ABC, AB=13, BC=14, and AC=15. Let D denote the midpoint of

¯

B

C

and

¯¯

with the bisector of angle BAC. Which of the

BC

let E denote the intersection of

¯

following is closest to the area of the triangle ADE?

(A)2 (B)2.5 (C)3 (D)3.5 (E)4

2. A round table has radius 4. Six rectangular place mats are placed on the table. Each

place mat has width 1 and length x as shown. Then are positioned so that each mat

has two corners on the edge of the table, these two corners being end points of the

same side of length x. Further, the mats are positioned so that the inner corners each

touch an inner corner of an adjacent mat. What is x?

(A)2 5 − 3

(B)3 (C)

3

7−

3

2

(D)2 3 (E)

5+2

3

2

3. Quadrilateral ABCD has AB=BC=CD, ∠ABC=70°, and ∠BCD=170°. What is the

degree measure of ∠BAD?

(A)75 (B)80 (C)85 (D)90 (E)95

¯¯

so that 2 ∙ BD =CD and ∠DAB=15°

BC

4. In △ABC, ∠ABC=45°. Point D is on

¯

.

Find ∠ACB.

(A)54° (B)60° (C)72° (D)75° (E)90°

5. Two particles move along the edges of equilateral triangle △ABC in the direction

A → B → C → A starting simultaneously and moving at the same speed. One starts at

¯¯

. The midpoint of the line segment A, and the other starts at the midpoint of

¯

B

C

joining the two particles traces out a path that encloses a region R. What is the ratio of

the area of R to the area of △ABC?

(A)

1

16

(B)

1

12

(C)

1

9

(D)

1

6

(E)

1

4

6. Isosceles △ABC has a right angle at C. Point P is inside △ABC, such that PA=11,

¯¯¯¯

PB=7, and PC=6. Legs

¯

A

C and

¯

BC have length s = a +b 2, where a and b are

positive integers. What is a +b ?

(A)85

(B)91 (C)108 (D)121 (E)127

¯¯¯¯¯

AB

, and C

D

bisects the

7. Triangle ABC has AC=3, BC=4, and AB=5. Point D is on

¯

right angle. The inscribed circles of △ADC and △BCD have radii r

a

and r

b

,

respectively. What is r

a

/r

b

?

(A)

1

28

5

(D)

56

10 − 2

10 −

2

(B)

10 − 2

56

3

3

(C)

31

14

10 −

2

(E)

28

10 −

2

8. Triangle ABC has ∠C=60° and BC=4. Point D is the midpoint of BC. What is the