Triangles Tutorial for SAT/GRE Test Preparation
|At least with triangles we are back to straight edges. This means that there is nothing tricky about the perimeter and area formulas. We will, however, need to spend some time after that to review angles, as well as the properties of some special triangles.|
|The perimeter of a triangle is the sum of the lengths of the sides.Perimeter = a + b + c
Cool fact: Any side of a triangle is always shorter than the sum of the other two sides. This is called the triangle inequality theorem. Think about it for a second. Imagine you have a ten foot piece of wood, and two four foot pieces of wood. Could you connect them at their ends to make a triangle? No, it just wouldn’t work. The four foot sections wouldn’t be long enough. And you could have known this before trying, because 10 > 4 + 4.
Triangle Inequality: For any triangle with sides a, b, c : a < b + c
|The area of a triangle is one half the length of the base times the height.Area = ½ Base × Height
Students often ask us where that formula comes from. The shaded region above is a rectangle with Height = h and Width = Base.
The area of that rectangle is Base × Height.
The triangle occupies exactly one half of the space of that rectangle, and so the area of the triangle is half the area of the rectangle.
|Okay, that about does it for perimeter and area. Let’s move on to some more facts about triangles.
The vertices of a triangle (or points where the lines meet) form angles. The sum of the measures of the angles within a triangle is 180 degrees.
Interesting fact: If side x of a triangle is opposite a larger angle than side y, then side x is longer than side y. Try drawing some triangles and check this out.
There are some special triangles to consider. If a triangle has two = sides, it is called an isosceles triangle. The angles opposite the two equal sides are equal (this follows from the interesting fact mentioned on the last page).If a triangle has three equal sides, it is an equilateral triangle, and all the angles measure 60 degrees.
The most famous family of triangles are the right triangles. Because of the right angle, it is easy to find the area of a right triangle. Just turn it so that the right angle forms the base and the height, and then apply the formula Area = ½ b × h. The Pythagorean theorem helps you solve for the third side of a right triangle when you know two of the other sides. The theorem says that
a2 + b2 = c2
|The 3-4-5 Right Triangle
And within the family of right-angled triangles, there are some even more special triangles.One of the nicest is the 3-4-5 triangle.
52 = 32 + 42
There are also the multiples of (3,4,5). That is, right triangles can come in the proportions (6,8,10) and ( 12,16,20) , etc.
Always study a question with a right angle carefully to see if it contains a 3-4-5 triangle, or a triangle derived from one (i.e. one whose dimensions are scaled).
|There are two other special right triangles. One is the isosceles right triangle, where the other two angles measures both equal 45 degrees. Here both these two angles and the corresponding opposite sides are =.In this triangle, the length of the hypotenuse equals the length of either leg times the square root of 2.
Finally there is the 30-60-90 triangle, where the sides can also be expressed as convenient proportions.
Keep an eye out for these triangles appearing either on their own, or as parts of more complicated geometrical figures.