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SAT数学知识讲解：三角形(1)

2013-08-09 16:33:54来源：互联网

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　　下面新东方在线为大家整理了SAT数学知识讲解：三角形(1)，供考生们参考，以下是详细内容。

　　Triangles pop up all over the Math section. There are questions specifically about triangles, questions that ask about triangles inscribed in polygons and circles, and questions about triangles in coordinate geometry.

　　Three Sides, Four Fundamental Properties

　　Every triangle, no matter how special, follows four main rules.

　　1. Sum of the Interior Angles

　　If you were trapped on a deserted island with tons of SAT questions about triangles, this is the one rule you’d need to know:

　　The sum of the interior angles of a triangle is 180°.

SAT数学知识讲解：三角形(1)

　　If you know the measures of two of a triangle’s angles, you’ll always be able to find the third by subtracting the sum of the first two from 180.

　　2. Measure of an Exterior Angle

　　The exterior angle of a triangle is always supplementary to the interior angle with which it shares a vertex and equal to the sum of the measures of the remote interior angles. An exterior angle of a triangle is the angle formed by extending one of the sides of the triangle past a vertex. In the image below, d is the exterior angle.

SAT数学知识讲解：三角形(1)

　　Since d and c together form a straight angle, they are supplementary:

　　. According to the first rule of triangles, the three angles of a triangle always add up to

　　, so

　　. Since

　　and

　　, d must equal a + b.

　　3. Triangle Inequality Rule

　　If triangles got together to write a declaration of independence, they’d have a tough time, since one of their defining rules would be this:

　　The length of any side of a triangle will always be less than the sum of the lengths of the other two sides and greater than the difference of the lengths of the other two sides.

　　There you have it: Triangles are unequal by definition.

　　Take a look at the figure below:

SAT数学知识讲解：三角形(1)

　　The triangle inequality rule says that c – b < a < c + b. The exact length of side a depends on the measure of the angle created by sides b and c. Witness this triangle:

SAT数学知识讲解：三角形(1)

　　Using the triangle inequality rule, you can tell that 9 – 4 < x < 9 + 4, or 5 < x < 13. The exact value of x depends on the measure of the angle opposite side x. If this angle is large (close to

　　) then x will be large (close to 13). If this angle is small (close to

　　), then x will be small (close to 5).

　　The triangle inequality rule means that if you know the length of two sides of any triangle, you will always know the range of possible side lengths for the third side. On some SAT triangle questions, that’s all you’ll need.

　　4. Proportionality of Triangles

　　Here’s the final fundamental triangle property. This one explains the relationships between the angles of a triangle and the lengths of the triangle’s sides.

　　In every triangle, the longest side is opposite the largest angle and the shortest side is opposite the smallest angle.

SAT数学知识讲解：三角形(1)

　　In this figure, side a is clearly the longest side and

　　is the largest angle. Meanwhile, side c is the shortest side and

　　is the smallest angle. So c < b < a and C < B < A. This proportionality of side lengths and angle measures holds true for all triangles.

　　See if you can use this rule to solve the question below:

SAT数学知识讲解：三角形(1)

　　What is one possible value of x if angle C < A < B?

　　(A)1

　　(B)6

　　(D)10

　　(E)15

　　According to the proportionality of triangles rule, the longest side of a triangle is opposite the largest angle. Likewise, the shortest side of a triangle is opposite the smallest angle. The largest angle in triangle ABC is

　　, which is opposite the side of length 8. The smallest angle in triangle ABC is

　　, which is opposite the side of length 6. This means that the third side, of length x, measures between 6 and 8 units in length. The only choice that fits the criteria is 7. C is the correct answer.

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