Wednesday, January 23, 2013

Right triangles: Starting with definitions

Nature does not create many straight lines, with the exception of some crystalline forms. Even rarer in nature are right angles, two straight lines meeting like a plus sign + to create four equal angles that add up to 360°, which means every angle created by the two crossing lines of the plus sign measures 90°.

Straight line geometry has been used for several  millennia by cultures around the world. When we teach geometry today, it is standard to teach the ideas discovered by the Greeks. For example, the symbol over by the angle on the left of this triangle is the lowercase Greek letter theta. It is a common convention to label angles with Greek letters, points with capital letters from the Roman alphabet and the lengths of straight lines with lowercase letters from the Roman alphabet.

In this picture, we have a right triangle. That little rectangle mark in the lower right hand corner signifies a 90° angle.

The side length opposite the right angle is called the hypotenuse. Since the three angles have a sum of 180°, the 90° angle accounts for half of that and the other two angles must add up to 90°. More than that, there is an easily proved property of triangles that the longest side of any triangle will be opposite the largest angle.  Sing the right angle must be the largest angle, the hypotenuse is always the longest side of a right triangle.

The other two sides are called the legs. If we discuss these legs, we often pick one of the smaller angles and distinguish the two legs based on their relation to the chosen smaller angle, which in this case is the one we labeled theta.

The leg that is not on of the line segments next to theta is called the opposite and the leg that creates the angle theta where it intersects with the hypotenuse is called the adjacent.

Tomorrow, we will discuss some of the relationships of the sides and angles of a right triangle using the terms associated with trigonometry.

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