Geometry: Size Matters, So Let's Measure
Size Matters, So Let's Measure
When I draw a triangle, one of the first things I do after I admire the color scheme and the straightness of the sides is compare my triangle to all other triangles around. Is my triangle the biggest one around? How can I decide what determines the biggest? For example, in Figure 11.8, which of the two triangles is the biggest?
One way to decide which triangle is bigger is to examine which triangle has the longest side. You could also base your decision on which triangle has the largest perimeter. Or you could base everything on the size of the enclosed area. This is a good time to discuss these properties.
Imagine that you have a triangular piece of property that you want to fence off. Before you head off to the lumber store to buy the materials, you will need to know how much fence to buy. You will need to know the lengths of all three sides of the region, and then you can add them together. When you measure the lengths of all three sides of a triangle and add them together, you are measuring the perimeter of the triangle.
The perimeter of a triangle is the sum of the lengths of the three sides.
The area of a triangle (or any other polygon) is a measurement of the amount of the region within an enclosed plane figure. It is a fairly simple task to calculate the area of a triangle. All that is needed is the length of its base and its height. The area of a triangle can be calculated by evaluating one-half the length of its base times its height.
If you measure the sides of your polygon in feet, then the area of the polygon will have units feet2 (read square feet). If the sides of our polygon are measured in yards, the units of area will be yards2 (read square yards). In general, if the length of a side is measured in “units,” the units of your area will be units2 (read square units).
But back to the nuts and bolts of calculating the area of a triangle, I need to explain what I mean by the base and the height. The height of a triangle is related to its altitude. An altitude of a triangle is a line segment drawn perpendicularly from a vertex of the triangle to the opposite side of the triangle. The length of the altitude is called the height of the triangle. The side opposite the vertex is called the base of the triangle.
A triangle's base and altitude depend on how it is oriented. Any side of a triangle can serve as its base. The side that is lowest in our figure is usually referred to as the base. Suppose you are dealing with ABC in Figure 11.9. In this case, ¯AB would be considered to be the base and ¯CD the altitude. The altitude does not have to lie in the interior of the triange, as is the case with RST, also shown in Figure 11.9.
The area of a polygon is the measurement of the size of the region within the polygon. The units of area are the units of length squared.
An altitude of a triangle is a line segment drawn perpendicularly from a vertex of the triangle to the opposite side of the triangle.
The height of a triangle is the length of its altitude.
The base of the triangle is the side opposite the vertex.
It is possible to calculate the area of ABC in Figure 11.9. All that has to be done is evaluate one-half the product of the length of its base and its height.
- Example 3: Find the area of ABC.
- Solution: The base of ABC has length 5, and its height is 8. The area of ABC is 12 × 5 × 8 = 20.
The area of a right triangle can be calculated by taking one-half the product of the lengths of the two legs.
Computing the area of a right triangle is fairly easy, if it has the right orientation. I recommend always using one leg for the base. Then, because the legs of a right triangle are perpendicular to each other, the other leg will serve as the altitude. So the area of a right triangle is one-half the product of the lengths of the legs. Figure 11.10 will shed some light on this situation.
Ladies and Gentlemen: The Pythagorean Theorem
The Pythagorean Theorem is one of the best known theorems in the world. There are several slick ways to prove this theorem. One method involves similar triangles (which you will learn about in Similar Triangles). Another technique makes use of the area of squares, which you will examine in Putting Quadrilaterals in the Forefront. As a theorem it is so useful that I can't wait to share it with you. Accept it as true for now and practice using it.
- Theorem 11.4: The Pythagorean Theorem. The square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the legs.
Practice using this theorem right now!
- Example 4: Given the right triangle shown in Figure 11.11, with AB = 5, and BC = 3, find the area of ABC.
- Solution: You are given that the hypotenuse of ABC has length 5, and that the length of one of the legs is 3. In order to find the area of a right triangle, all you need to know are the lengths of the two legs. You are given the length of one leg and the hypotenuse. You need the length of the second leg. The Pythagorean Theorem is here to save the day.
- (AC)2 + (BC)2 = (AB)2
- (AC)2 + 32 = 52
- (AC)2 = 25 - 9 = 16
- AC = 4
- Now that you know the length of both legs, you can find the area:
- Area = 12(AC) × (BC) = 12(4)(3) = 6. So the area of ABC is 6.
The Triangle Inequality
Sometimes things in life are just not equal. Your neighbor might have more money than you, or you might be smarter than some of your friends. As much as mathematicians would like everything to be equal in the world, it can't be. Inequality exists, and there's nothing that can be done about it. Some numbers are bigger than others. And some aspects of geometric figures are unequal as well. The Triangle Inequality is a theorem about the inequity of the sides of a triangle.
- Theorem 11.5: Triangle Inequality. The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
You can write a formal proof of the triangle inequality. Figure 11.12 shows
Prove: AB + AC > BC
Proof: You need a game plan. If you drop a perpendicular segment from A to ¯AB, you can use the Pythagorean Theorem to show that AB > BD and AC > DC (see Figure 11.13). Whenever you add some detail to your drawing, such as introducing perpendicular lines, your justification is “construction.” You are allowed to construct things like parallel and perpendicular lines by Euclid's Parallel Postulate and Theorem 10.1. The Additive Property of Inequality shows that AB + AC > BD + DC. If you apply the Segment Addition Postulate to show that BD + DC = BC, you are done. Bring out the columns!
|2.||Draw ¯AD ¯BC||Construction (or Theorem 10.1)|
|3.||(BD)2 + (AD)2 = (AB)2 and (CD)2 + (AD)2||The Pythagorean Theorem|
|4.||(AB)2 > (BD)2 and (AC)2 > (DC)2||Definition of >|
|5.||AB > BD and AC > DC||Algebra|
|6.||AB + AC > BD + DC||Additive Property of Inequality|
|7.||BD + DC = BC||Segment Addition Postulate|
|8.||AB + AC > BC||Substitution (steps 6 and 7)|
You can use the Triangle Inequality to determine if triangles with certain dimensions can exist.
- Example 5: Can a triangle have sides with length 3, 4, and 7?
- Solution: By the Triangle Inequality, the length of any one side must be less than the sum of the lengths of the other two sides. Because 3 + 4 = 7, this triangle violates the Triangle Inequality and cannot exist.
Here's your chance to shine. Remember that I am with you in spirit and have provided the answers to these questions in Answer Key.
- 1. What would you call a triangle with two sides congruent and one obtuse angle?
- 2. Prove that the acute angles of a right triangle are complementary.
- 3. Write an informal (two-column proof) of Theorem 11.3.
- 4. Find the area of RST in Figure 1.9.
- 5. Find the area of RST shown in Figure 11.14, if RS = 13 and ST = 5.
- 6. Can a triangle have sides of length 3, 4, and 8?
Excerpted from The Complete Idiot's Guide to Geometry © 2004 by Denise Szecsei, Ph.D.. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with Alpha Books, a member of Penguin Group (USA) Inc.