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第四章:部分分式 Partial Fraction

第六章:角的形成及单位 Angles and Measurements

第十三章: 方程组 Simultaneous Equations

第十五章:二元一次不等式及线性规划 linear inequality in two variables and linear programming

11.3 三角形的面积 Area of Triangles

11.3 Area of Triangles 三角形的面积


 

1. Given A(4p, 1), B(2, 5) and C(-2p, 3), find the possible value of p if

已知A(4p, 1), B(2, 5) and C(-2p, 3), 求 p 的可能值若

(a) the area of triangle ABC is 14 unit^2,

三角形ABC 的面积是14单位^2

(b) A, B and C are collinear.

A, B 和 C 共线

Answer:
(a) 1.5, –2

(b) -\frac{1}{4}

2. Given that A(-4, -3), B(2, 0) and C(8, -8) are the vertices of a triangle.

已知A(-4, -3), B(2, 0) 和 C(8, -8) 是一三角形的顶点。

(a) Calculate the area of triangle ABC,

计算三角形ABC 的面积

(b) Find the length of AC and hence calculate the perpendicular distance from B to AC.

求AC 的长及据此计算从B 到AC 的垂直距离。

Answer:
(a) 33unit^2

(b) 13, \frac{66}{13}

3. The coordinates of the vertices of a triangle are (3a, a), (2, 3) and (4, 7) in anticlockwise direction. Find the area of the triangle in terms of a. If the area of the triangle is 16 unit^2, find the value of a.

一三角形的顶点坐标按逆时顺序是 (3a, a), (2, 3) 和 (4, 7). 用 a 表示求三角形的面积. 若三角形的面积是 16 单位^2, 求 a的值.

Answer:
1 – 5a, –3
4. Given that A, B and C are the three points with coordinates (-3, 8), (2, 3) and (-1, k) respectively where k > 0.
Get the statement of the area of triangle ABC in terms of k. Hence, find the value of k when

已知A, B 和 C 的坐标分别是(-3, 8), (2, 3) 和 (-1, k) 其中 k > 0.
用k表示求三角形ABC 面积的式子。据此,求k 的值当

(a) the area of triangle ABC is 25 unit^2,

三角形ABC 的面积是25单位^2

(b) A, B and C are on the collinear.

A, B 和C 共线。

Answer:
(a) 16, -4

(b) 6

5. Find the area of the triangle, whose vertices are (2, -1), (t + 1, t – 3), (t + 2, t). Hence show that when t = \frac{1}{2}, these points are collinear.

求顶点为 (2, -1), (t + 1, t – 3), (t + 2, t) 的三角形面积. 据此, 证当t = \frac{1}{2} 时, 这些点会共线.

Answer:
\frac{1}{2}|2t - 1|
6. Given three vertices of a triangle ABC, A(-2, 2), B(3, 7) and C(4, 0). If the area of \bigtriangleupACD and \bigtriangleupABC are the same and \bigtriangleupACD = 90^{\circ}, find the coordinate of D.

已知一三角形的三个顶点, A(-2, 2), B(3, 7) 和 C(4, 0). 若 \bigtriangleupACD和 \bigtriangleupABC 面积相同及 \bigtriangleupACD = 90^{\circ}, 求D 的坐标.

Answer:
(6, 6), (2, -6)
7. Given that O, A, B are the points (0,0), (1, 2), (4, 0), and C lies on the line x = 2. If the areas of \bigtriangleupOAB and \bigtriangleupOBC are equal, find the coordinates of C.
已知O, A, B 为点(0,0), (1, 2), (4, 0), 及C 在直线x = 2上. 若\bigtriangleupOAB 和 \bigtriangleupOBC的面积相同, 求C的座标.
Answer:
(2, 2), (2, -2)