第四章:部分分式 Partial Fraction

第六章:角的形成及单位 Angles and Measurements

8.5 三角形的外接圆半径和内切圆半径 Radius of Circumscribe Circle and Inscribe Circle of Triangles [练习题]

(1)
If the angles of a triangle are in a ratio 4 : 5 : 6, and the perimeter of the triangle is 100cm, find
若三角形的角的比例是 4 : 5 : 6, 且三角形的周长是 100cm, 求

(a) the radius of the circumscribe circle.
外接圆的半径,

(b) the greatest side of the triangle.
三角形的最大边。

答案 Answer:

(a) 19.48cm

(b) 37.11cm

(2)
Given that (a + b) : (b + c) : (c + a) = 10 : 15 : 11 and the area of the triangle ABC is \frac{3}{2}\sart{3}cm^2. Calculate
已知(a + b) : (b + c) : (c + a) = 10 : 15 : 11 及三角形的面积是 \frac{3}{2}\sart{3}cm^2. . 计算

(a) the length of the largest side,
最大的边长,

(b) the radius of inscribe circle,
内切圆的半径

(c) the radius of circumscribe circle.
外接圆的半径

答案 Answer:

(a) 4cm
(b) \frac{\sqrt3}{3}cm
(c) 2.02cm

(3)
B = 60^{\circ}, b = 4, area = \sqrt{3} , find求

(a) a, c, (a > c)

(b) the radius of the inscribe circle,
内切圆半径,

(c) the radius of the circumscribe circle.
外接圆半径。

答案 Answer:

(a) 4.38, 0.91                     

(b)  0.37                       

(c)   2.31

(4)


In Fig, the circle O is the circumscribe circle of \bigtriangleupABC as well as the inscribe circle of \bigtriangleupPQR. If PQ = 10cm, QR = 12cm, RP = 14cm, find

图中,圆O 同时是 \bigtriangleupABC 的外接圆以及 \bigtriangleupPQR 的内接圆. 若PQ = 10cm, QR = 12cm, RP = 14cm, 求

 

(a) the area of \bigtriangleupPQR
\bigtriangleupPQR 的面积

(b) the radius of the circle O
圆O 的半径

(c) \angleP

(d) the length of the side AB.
AB 的边长。

答案 Answer:

(a) 24\sqrt{6}cm^2

(b) 3.27cm

(c) 57.12^{\circ}

(d) 5.744cm

(5)
Let the three sides of a triangle be a, b, c and A is a angle opposite to the side a.
If (b + c) : (c + a) : (a + b) = 6 : 4 : 5, and perimeter of \bigtriangleupABC is , find

已知a, b, c 为一三角形的三边和 A 是 a 边所对的角. 若 (b + c) : (c + a) : (a + b) = 6 : 4 : 5, 及 ∆ABC 的周长为 , 求

(a) sinA : sinB : sinC,

(b) \angleA,

(c) the area of \bigtriangleupABC,
\bigtriangleupABC 的面积,

(d) radius of circumscribe circle of \bigtriangleupABC
\bigtriangleupABC 的外接圆半径,

(e) radius of inscribe circle of \bigtriangleupABC.
\bigtriangleupABC 的内切圆半径。

答案 Answer:
(a) 3 : 7 : 5

(b) 21.78^{\circ}

(c) 19.48 单位^2

(d) 7

(e) 1.5

(6)
Given that A, B and C are on circle O where curve AB : curve BC : curve AC = 3 : 4 : 5, the length AB is 10\sqrt{2} , then calculate

已知A, B 和 C 在圆O 上且弧 AB : 弧 BC : 弧 AC = 3 : 4 : 5, AB 边长是10\sqrt{2} , 计算

(a) the radius of circle O,
圆O 的半径,

(b) the area of \bigtriangleupABC,
\bigtriangleupABC 的面积

(c) the radius of the inscribe circle of \bigtriangleupABC.
\bigtriangleupABC 的内切圆半径

答案 Answer:
(a) 10
(b) 118.3
(c) 4.66
(7)

Given that s is a semi-perimeter of \bigtriangleupABC and (s – a) : (s – b) : (s – c) = 7 : 2 : 1. Find the radius of the inscribe circle if the area of \bigtriangleupABC is 8\sqrt{35}cm^2.

已知s 为 \bigtriangleupABC 的半周长且(s – a) : (s – b) : (s – c) = 7 : 2 : 1. 若\bigtriangleupABC 的面积是8\sqrt{35} cm^2求内切圆半径。

答案 Answer:
\frac{2}{5}\sqrt{35}cm