Triangles(5)

　　.　　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 TriangleThe 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 . 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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