Thursday, June 28, 2012

Classifications of triangles

There are two ways to classify triangles. One is by looking at the largest of the three angles.

Obtuse: The largest angle is greater than 90°.
Right: The largest angle is exactly 90°.
Acute: The largest angle is less than 90°.

Make sure that when you are looking at the angles, you only consider the largest angle.  Every triangle must have at least two acute angles, but we are not interested in all the angles in this classification, only the largest one.

The other classification system deals with the relationships between the angles.

Equilateral: In an equilateral triangle, all three angles are equal.  Since they have to add up to 180°, the angle are 60°, 60° and 60°.
Isosceles: At least two of the angles are equal.  Technically, an equilateral triangle is a special case of isosceles, but isosceles also includes 70°-70°-40°, 40°-40°-100°, 32.2°-32.2°-113.6° and any of an infinite set of three angles where two of the angles are the same.
Scalene: All three angles are different.

If instead of giving the three angles of a triangle, a triangle can be defined by giving the three side lengths.  Not any three positive numbers can be the lengths of the sides of a triangle, because they must conform to the triangle inequality. The simplest way to say it in English is that the two short sides must add up to at least the equal of the long side.  If we call the side lengths r, s and t, where t is the longest, if r + s = t, the drawing would not be of a triangle, but instead a line segment with a point on the line segment that should be the apex of the triangle.  In class, I called this the degenerate case. As a student noted, we can think of it as a "squished triangle".

The easiest formula for the triangle inequality does not force us to find the long side. The formula is

r + s >= t >= | r - s |

The straight line brackets indicate absolute value. The editor for the blog does not have the "greater than or equal to" sign, so I have to type >= to signify this.

Example: 15, 6, 7 are NOT the sides of a triangle. The simplest way to state why is that 6+7 < 15, but I could also try to plug the numbers into the triangle inequality in any order

15 + 6 > 7.  This part is fine.
7 < |15 - 6| = 9 This is where the inequality breaks down, because 7 is less than 9, the difference between 15 and 6.

If the three lengths r, s, t can be the sides of a triangle, both parts of the inequality will be true.  If the lengths don't work, one of the inequalities will fail. If we have three lengths that do make a legitimate triangle, we can figure out the classifications as follows.

Equilateral:  All three sides are the same length.
Isosceles: At least two side lengths are the same.
Scalene: All three side lengths are different.

For the other classification system, we rely on the Pythagorean Theorem.  Here, we need to identify the longest side of the three.  We are used to using a, b and c for the lengths in the Pythagorean Theorem, where c is the longest side.

If a² + b² = c², we have a right triangle.
If a² + b² > c², we have an acute triangle.
If a² + b² < c², we have an obtuse triangle.

Here is a link to all the classification of triangles posts, which will also include practice problems for other ideas including Heron's formula.

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