Download 2013年1月12日托福写作真题回忆

SAT 数学的三角形知识讲解二
SAT 数学考试中的三角形知识有哪些？SAT 资料下载的小编为考生们整理了这些知识，让我们一起来学
习一下吧！
The guy who named 30-60-90 triangles didn’t have much of an imagination. These triangles
have angles of
,
, and
. What’s so special about that? This: The side lengths of 30-60-90 triangles always follow
a specific pattern. Suppose the short leg, opposite the 30° angle, has length x. Then the
hypotenuse has length 2x, and the long leg, opposite the 60° angle, has length x
. The sides of every 30-60-90 triangle will follow this ratio of 1:
: 2 .
全国免费咨询电话：400-0123-267
This constant ratio means that if you know the length of just one side in the triangle,
you’ll immediately be able to calculate the lengths of all the sides. If, for example, you
know that the side opposite the 30º angle is 2meters long, then by using the 1:
: 2 ratio, you can work out that the hypotenuse is 4 meters long, and the leg opposite the 60º
angle is 2
meters.
And there’s another amazing thing about 30-60-90 triangles. Two of these triangles joined
at the side opposite the 60º angle will form an equilateral triangle.
全国免费咨询电话：400-0123-267
Here’s why you need to pay attention to this extra-special feature of30-60-90 triangles.
If you know the side length of an equilateral triangle, you can figure out the triangle’s height:
Divide the side length by two and multiply it by
. Similarly, if you drop a “perpendicular bisector” (this is the term the SAT uses) from any
vertex of an equilateral triangle to the base on the far side, you’ll have cut that triangle
into two 30-60-90triangles.
Knowing how equilateral and 30-60-90 triangles relate is incredibly helpful on triangle,
polygon, and even solids questions on the SAT. Quite often, you’ll be able to break down these
large shapes into a number of special triangles, and then you can use the side ratios to figure
out whatever you need to know.
45-45-90 Triangles
A 45-45-90 triangle is a triangle with two angles of 45° and one right angle. It’s sometimes
called an isosceles right triangle, since it’s both isosceles and right. Like the 30-60-90
triangle, the lengths of the sides of a 45-45-90 triangle also follow a specific pattern. If
the legs are of lengthx (the legs will always be equal), then the hypotenuse has length x
:
Know this 1: 1:
ratio for 45-45-90 triangles. It will save you time and may even save your butt.
全国免费咨询电话：400-0123-267
Also, just as two 30-60-90 triangles form an equilateral triangles, two45-45-90 triangles
form a square. We explain the colossal importance of this fact when we cover polygons a little
later in this chapter.
Similar Triangles
Similar triangles have the same shape but not necessarily the same size. Or, if you prefer
more math-geek jargon, two triangles are “similar” if the ratio of the lengths of their
corresponding sides is constant (which you now know means that their corresponding angles must
be congruent). Take a look at a few similar triangles:
As you may have assumed from the figure above, the symbol for “is similar to” is ~. So,
if triangle ABC is similar to triangle DEF, we writeABC ~ DEF.
There are two crucial facts about similar triangles.
Corresponding angles of similar triangles are identical.
Corresponding sides of similar triangles are proportional.
For ABC ~ DEF, the corresponding angles are
The corresponding sides are AB/DE = BC/EF = CA/FD.
全国免费咨询电话：400-0123-267
The SAT usually tests similarity by presenting you with a single triangle that contains
a line segment parallel to one base. This line segment creates a second, smaller, similar triangle.
In the figure below, for example, line segment DE is parallel to CB, and triangle ABC is similar
to triangle AE.
After presenting you with a diagram like the one above, the SAT will ask a question like
this:
If
= 6 and
=
, what is
?
Notice that this question doesn’t tell you outright that DE and CB are parallel. But it
does tell you that both lines form the same angle, xº, when they intersect with BA, so you should
be able to figure out that they’re parallel. And once you see that they’re parallel, you should
全国免费咨询电话：400-0123-267
immediately recognize that ABC ~ AED and that the corresponding sides of the two triangles are
in constant proportion. The question tells you what this proportion is when it tells you that
AD = 2 /3AC. To solve for DE, plug it into the proportion along with CB:
Congruent Triangles
Congruent triangles are identical. Some SAT questions may state directly that two triangles
are congruent. Others may include congruent triangles without explicit mention, however.
Two triangles are congruent if they meet any of the following criteria:
All the corresponding sides of the two triangles are equal. This is known as the Side-Side-Side
(SSS) method of determining congruency.
The corresponding sides of each triangle are equal, and the mutual angles between those
corresponding sides are also equal. This is known as the Side-Angle-Side (SAS) method of
determining congruency
全国免费咨询电话：400-0123-267
.
The two triangles share two equal corresponding angles and also share any pair of corresponding
sides. This is known as the Angle-Side-Angle (ASA) method of determining congruency
.
Perimeter of a Triangle
The perimeter of a triangle is equal to the sum of the lengths of the triangle’s three
sides. If a triangle has sides of lengths 4, 6, and 9, then its perimeter is 4 + 6 + 9 = 19.
Easy. Done and done.
Area of a Triangle
The formula for the area of a triangle is
where b is the length of a base of the triangle, and h is height (also called the altitude).
The heights of a few triangles are pictured below with their altitudes drawn in as dotted lines.
全国免费咨询电话：400-0123-267
We said “a base” above instead of “the base” because you can actually use any of the
three sides of the triangle as the base; a triangle has no particular side that has to be the
base. You get to choose.
The SAT may test the area of a triangle in a few ways. It might just tell you the altitude
and the length of the base, in which case you could just plug the numbers into the formula.
But you probably won’t get such an easy question. It’s more likely that you’ll have to find
the altitude, using other tools and techniques from plane geometry. For example, try to find
the area of the triangle below:
To find the area of this triangle, draw in the altitude from the base (of length 9) to the
opposite vertex. Notice that now you have two triangles, and one of them (the smaller one on
the right) is a 30-60-90 triangle.
全国免费咨询电话：400-0123-267
The hypotenuse of this 30-60-90 triangle is 4, so according to the ratio1:
: 2, the short side must be 2 and the medium side, which is also the altitude of the original
triangle, is 2
. Now you can plug the base and altitude into the formula to find the area of the original triangle:
1/ 2bh =1/2(9)(2
) = 9
.
Trig or Treat?
“The new SAT includes trigonometry? Yikes!” If you’ve heard people talking this particular
kind of jive, don’t listen to it. The people freaking out don’t know anything about the test.
Here’s what the actual SAT people say about trig questions on the new SAT: “These questions
can be answered by using trigonometric methods, but may also be answered using other methods.”
You will never have to use trig to solve a problem, and we’ll come right out and say it: You
never should use trig. That’s right. We’ll even quote us on that: “You never should use trig.”
The questions on which you could (but shouldn’t) use trig on the new SAT will cover 30-60-90
and 45-45-90 triangles. And the methods you already learned in this book for dealing with those
triangles are faster and easier than using trig. So forget trig.
来源于：小马过河
小马过河资料下载频道，欢迎您来下载！
全国免费咨询电话：400-0123-267