Notes for the week of October 12 through 15

Notes for area of a triangle defines by three points and the distance formula.

Notes for interior angles sums of polygons and the measure of regular angles.

More practice on the interior angles sums and regular angle measurements.

Classifying triangles defined by three vertices in the plane.

In class, we have worked with three points in the plane where one is the origin (0, 0) and the other two are labeled (a, b) and (c, d).  In the notes here on the blog, the points are given coordinates with subscripts such as (x1, y1) and (x2, y2). The formulas are the same, only the letters used in the formulas are different.

The area for the triangle define by (0, 0), (a, b) and (c, d) is given by the formula


Area = ½| ad - bc |

This formula is relatively simple compared to Heron's formula, with no square roots necessary. On the other hand, we will need to do three square roots to get the lengths of the three line segments for both classification methods.  Let's look at an example.

Example # 1: (0, 0), (6, 3), (7, 1)

Area: ½| (6)(1) - (3)(7)| = ½| 6 - 21| = ½| - 15 | = ½(15) = 7.5

Distances:

Between (0, 0) and (6, 3): When (0, 0) is involved the subtractions are easy and we get

Distance = sqrt((0-6)² + (0-3)²) = sqrt(6² + 3²) = sqrt(36 + 9) = sqrt(45)

Usually, I'd ask to have this simplified, but this problem just leave the answer as the square root of 45, it will make the next steps easier.

Between (0, 0) and (7, 1) 

Distance = sqrt((0-7)² + (0-1)²) = sqrt(7² + 1²) = sqrt(49 + 1) = sqrt(50)

Between (6, 3) and (7, 1)   

Distance = sqrt((6-7)² + (3-1)²) = sqrt((-1)² + 2²) = sqrt(1 + 4) = sqrt(5)

Now we have the distances, the classifications can be done.

Equilateral, isosceles or scalene?  The numbers are all different sqrt(45), sqrt(50) and sqrt(5), so the triangle is scalene.

Obtuse, right or acute? Here we need to square the distances, which is easy because they are all written as square roots, so the squares are 45, 50 and 5.  We need to compare the sum of the two short sides squared versus the long side squared and in this case

45 + 5 = 50.

This is a right triangle.

Final answer: Area of 7.5 square units, a scalene right triangle.

Example #2: (3, 2), (4, 1), (1, 7)

Here, we don't have a point at the origin, so we will subtract a value of one given vertex from all of the vertices.  That will give us three new points, but the triangle defined by our new points will have the same area and classifications.  I will choose to subtract (3, 2), but using (4, 1) or (1, 7) would also work.

New vertex #1: (3, 2) - (3, 2) = (0, 0)
New vertex #2: (4, 1) - (3, 2) = (1, -1)
New vertex #4: (1, 7) - (3, 2) = (-2, 5)

Area = ½| (1)(5) - (-1)(-2)| = ½| 5 - 2| = ½| 3 | = ½(3) = 1.5


Between (0, 0) and (1, -1):

Distance = sqrt((0-1)² + (0-(-1))²) = sqrt(1² + 1²) = sqrt(1 + 1) = sqrt(2)

Usually, I'd ask to have this simplified, but this problem just leave the answer as the square root of 45, it will make the next steps easier.

Between (0, 0) and (-2, 5) 

Distance = sqrt((0-(-2))² + (0-5)²) = sqrt(2² + 5²) = sqrt(4 + 25) = sqrt(29)

Between (1, -1) and (-2, 5)   

Distance = sqrt((1-(-2))² + ((-1)-5)²) = sqrt(3² + (-6)²) = sqrt(9 + 36) = sqrt(45)

Now we have the distances, the classifications can be done.

Equilateral, isosceles or scalene?  The numbers are all different sqrt(2), sqrt(29) and sqrt(45), so the triangle is scalene.

Obtuse, right or acute? Here we need to square the distances, which is easy because they are all written as square roots, so the squares are 2, 29 and 45.  We need to compare the sum of the two short sides squared versus the long side squared and in this case

2 + 29 = 31 < 45.

This is an obtuse triangle.


Final answer: Area of 1.5 square units, a scalene obtuse triangle.

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.

The triangle inequality, Heron's formula and the test for acute, right or obtuse using side lengths.

You probably already learned in high school that the interior angles of a triangle always add up to 180°.  Just as importantly, the Triangle Inequality tells us that any two side lengths of a triangle must add up to more than the third side length.  Another way to say this is that every side length must be less than half the total perimeter.

The idea of half the perimeter or semi perimeter shows up in Heron's Formula, a way to find the area of a triangle if you are given the three side lengths.

We also have a test using side lengths to see if a triangle is acute, right or obtuse.  If we have three side u, v and w and we declare that w is the long side, then we can use the sum of the squares of the short side to see how a triangle is classified.

u² + v² < w² : The triangle is obtuse


u² + v² = w² : The triangle is right

u² + v² > w² : The triangle is acute

Here are some practice problems for area and classification.

For each of these triples of numbers:
1) Determine if the triangle is equilateral, isosceles or scalene.
2) Determine if the triangle is acute, right or obtuse.
3) Find the area as a simplified square root
4) Find the area rounded to the nearest thousandth

a) 1, 1, 1
b) 1, 2, 2
c) 1, 3, 3
d) 1, 4, 4
e) 2, 2, 2
f) 2, 2, 3
g) 2, 3, 3
h) 2, 3, 4
i) 2, 4, 4
j) 3, 3, 3
k) 3, 3, 4
L) 3, 4, 4
m) 4, 4, 4

Answers in the comments.

practice for the take home and in class triangle identification problems

Part A) In all the following problems where triangles are defined by lengths, we will use the number 11 and the other numbers must be whole numbers.

obtuse and isosceles: 11, 11, _____
acute and isosceles: 11, 11, _____
obtuse and scalene: 11, ___, ___
right and scalene: 11, ____, ____ (hint: 11 will be one of the legs.)
acute and scalene: 11, ____, ____


Part B) In all the following problems where triangles are defined by angles, we will use one angle of 34° and the other numbers must be whole numbers.

obtuse and isosceles: 34°, ____, _____
acute and isosceles: 34°, ____, _____
obtuse and scalene: 34°, ___, _____ (many correct answers.)
right and scalene: 34°, ____, ____
acute and scalene: 34°, ____, ____ (many correct answers.)

Answers in the comments.

practice for triangles, angles and sides

In this picture, we are told the measure of three angles. This is enough information to find the measure of all the angles.

1) Find the measure of angles DCA, DAC and CBA.

There is also a rule in a triangle that if one angle is larger than another, the opposite side of the big angle must be longer than the opposite side of the smaller angle.

2) Use the rule to put this list of line segments in order from longest to shortest. AB, AC, AD, BC, CD.

Answers in the comments.

Classifying triangles

There are two ways to classify triangles. The first way deals with the size of the largest angle in the triangle.

Obtuse triangle: A triangle is obtuse if the largest angle is greater than 90 degrees in measure. If the sides are given as a, b and c, where c is the longest side, if a^2 + b^2 < c^2, the triangle is obtuse.

Right triangle: If the largest angle is exactly 90 degrees in measure, the triangle is a right triangle. If the sides are given as a, b and c, where c is the longest side, a^2 + b^2 = c^2 means we have a right triangle. This is known as the Pythagorean Theorem.

Acute triangle: If the largest angle is less than 90 degrees, the triangle is called an acute triangle. If the sides are given as a, b and c, where c is the longest side, a^2 + b^2 > c^2 means we have an acute triangle.

The other method of classification has to do with having angles of the same measure or sides of the same length.

Equilateral triangle: All three angles have the same measure, 60 degrees. All three sides are the same length.

Isosceles triangle: At least two sides have the same measure, which also means two angles must have the same measure. Technically, and equilateral triangle is still isosceles, but it is a special case and we usually name it by the more specific name.

Scalene triangle: All three sides are different lengths, and all three angles are different measure.


Practice problems.

1) Can a triangle be both a right triangle and an isosceles triangle?

2) A triangle has side lengths 3, 3 and 4. What are its two classifications?

3) A triangle has two angles, 70 degrees and 15 degrees. What are its two classifications?

4) An isosceles triangle has one angle of measure 40 degrees. What are the two possible sets of angle measures and classify both as acute, right or obtuse.

5) A triangle has side lengths 5, 12 and 13. What are its two classifications?

Answers in the comments.