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SAT数学:Triangles三角形知识讲解(2).

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30-60-90 Triangles 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 .

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 2 meters 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.

Here’s why you need to pay attention to this extra-special feature of 30-60-90 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 30-60-90 triangles. 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 length x (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. Also, just as two 30-60-90 triangles form an equilateral triangles, two 45-45-90 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 prer 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 write ABC ~ 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. 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 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 .
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.

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.

The hypotenuse of this 30-60-90 triangle is 4, so according to the ratio 1: : 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.

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