在GRE数学几何知识点中，三角形的知识是非常重要的。下面，小编就为大家实例讲解GRE数学知识点之三角形，帮助大家巩固这个数学知识点。
1.三角形的基本性质
① Length of sides： In any triangle, each side is shorter than the sum of the lengths of the other two sides(在三角形中，任一边的长度小于其他两边长度的和)。 推论：三角形中两边之差小于第三边。
② Angle measures： In any triangle，the sum of the three interior angles is 180°.
③ Angles and opposite sides： In any triangle，the relative angle sizes correspond to the relative lengths of the sides opposite those angles. In other words，the smaller the angle，the smaller the side opposite the angle (and vice-versa). Accordingly，if two angles are equal in size, the sides opposite those angles are of equal length (and vice-versa).
推论：
I.三角形中若最小的两条边的平方和小于第三条边的平方和，则此三角形必为钝角三角形。
II.三角形中若最小的两条边的平方和大于第三条边的平方和，则此三角形必为锐角三角形。
④ Area of a triangle：The area of any triangle is equal to 1/2 the product of its base and its height (height is also called the altitude)：
Area= 1/2 • base• altitude(height) = 1/2bh
在知道三角形的三边之长的情况下，可以用一特殊公式来求解三角形的面积。设三角形的三边边长分 别为a，b, c，s = (a+b+c)/2，则三角形面积为: SΔ = √s(s— a) (s— b) (s — c)
⑤若两个三角形相似，则这两三角形的面积比等于相似比的平方。
⑥三角形的一个外角等于其不相邻的两个内角之和。
2. Right Triangles(直角三角形)
勾股定理：a2 +b2 = c2
3. Special Right Triangles(特殊的直角三角形)
In two ( and only two) of the unique triangles we’ve identified as Pythagorean triplets, all degree measures are integers(在我们所定义的毕达哥拉斯三角形中，仅有两种所有的角的度数都是整数的三 角形)。
① The corresponding angles opposite the sides of a 1 : 1 : √2 triangle are 45°，45°, and 90°.
② The corresponding angles opposite the sides of a 1 : √2 : 2 triangle are 30°，60°，and 90°.
注：Two 45°— 45°— 90°triangles pieced together form a square, and two 30°—60°—90°triangles together form an equilateral triangle.
4. Isosceles Triangles (等腰三角形)
等腰三角形具有以下性质：
① Two of the sides are congruent (equal in length).
③ A line that bisects the angle formed by the equal sides bisects the opposite side.
5. Equilateral Triangles(等边三角形)
等边三角形具有以下性质：
① All three sides are congruent (equal in length)
② All three angles are 60°
③ The area=s2√3/4(s=the length of one side)
④ Any line bisecting one of the 60° angles divides an equilateral triangle into two right triangles with angle measures of 30°，60°，and 90°; in other words, into two 1 : √3 : 2 triangles.
GRE数学知识点三角形实例分析
A The area of the shaded region
B The sum of the areas of the two unshaded triangular regions
本题的正确答案为(A)。阴影区的面积=1/2 I AD | h,非阴影区面积和=1/2 I BC | h，因为BC与A夹角为89°,所以|AD|>|BC|，由此可得阴影区面积>两个非阴影三角形面积的和。
Triangular garden ABC is redesigned by increasing the length of AC by 20 percent to point C’ and decreasing the length of AB by 20 percent to point B’.
A The area of the original garden ABC
B The area of the redesigned garden AB’C’
题干的意思是：重新设计三角形花园ABC,使其AC边长增加20%到C’点;边长减少20%到B’点。
解：本题的正确答案为(A)。从表面上看，好像是变化前后花园的面积大小不变，但是若考生根据题意列出方程就可发现，变化后花园的面积是减小了：S△A’B’C’=1/2|AB’|•|AC|= 1/2(1-20%)|AB| • (1 + 20%) |AC| =0. 96 S△ABC
以上就是小编为大家整理的GRE数学知识点之三角形实例讲解，希望对大家有一定的帮助。
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