CHAPTER 3
METHODOLOGY
The study on the Conditions of Triangles having areas and perimeters numerically equal is an extension to the invitation of Jean E. Kilmer, West Covina, California and Perfect Triangles of Jim Wilson, EMT 724, University of Georgia such triangles with integer sides, integer area, and numerically equal area and perimeter and that they said: 1. Find all such triangles, 2. What right triangles with integer sides are perfect triangles? 3. Given that DABC has equal area and perimeter, show that the radius of circle p is xy/2.
Since then, the researcher developed an interest to extend further the geometric and algebraic properties of the triangle circumscribing a circle of radius 2.
To be acquainted with existing geometric and algebraic properties of triangles and circles, the researcher read different books of well known mathematicians whose work were related to triangle with equal area and perimeter.
Based on the mathematical ideas among the mathematicians, the researcher extended the concept to when a triangle have area and perimeter numerically equal.
The researcher made geometric representations, such as drawings of lines, triangle circumscribing a circle to generate conditions on triangles having areas and perimeters numerically equal. Requirements were determined that the sides had to meet and that the tangent segments had to meet before a triangle of equal area and perimeter could be found. Examples of triangles having equal areas and perimeters were given. Finally, the statements and conjectures were clearly defined and proven to be true and that resulted to a new theorem.
The concept pertaining to the characteristics and properties of triangles should be considered in order to formulate different theorems particularly, to the relationship of its area and perimeter. The researcher presented the different attributes of triangles and circles. These attributes (i.e. angles, sides, circumference, types, etc.) were presented in basic manner in order to devise easy to understand theorems. This study focuses to the benefits of the students and instructors who are fun of investigating and finding proofs on geometric construction specifically, the geometric construction that involves lines, triangles and inscribed circle.
Triangles
Triangles, as the prefix tri suggests, are closed geometrical figures that have three straight sides. Every triangle will, as a result, have three angles as well. A triangle is one of the basic shapes of geometry: a two-dimensional figure with three vertices and three sides which are straight line segments.
One important thing to remember about triangles is that the sum of the three angles in a triangle is always equal to 180 degrees.
Types of Triangles
Special relationships between the three sides and the three angles allow us to define types of triangles that are nicer to work with and understand, at least at a basic starting level. Triangles can be classified according to their side lengths: a triangle is called
· equilateral if all its sides have the same length (or equivalently: all its angles are equal)
· isosceles if (at least) two of its sides have the same length (or equivalently: two of its angles are equal)
· scalene if all its sides have different lengths (or equivalently: all its angles are different)
Triangles can also be classified according to the size of their largest angle: a triangle is called
· right if one of its angles is a right angle (90 degrees or π/2 radians). The side opposite the right angle is called the hypotenuse. It is the longest side in the right triangle.
· obtuse if one angle is bigger than a right one
· acute if each angle is smaller than a right one
Basic facts
A triangle is a polygon and a 2-simplex . Two triangles are said to be similar if one can be produced by uniformly expanding the other. In this case, the lengths of their sides are proportional. That is, if the longest side of a triangle is twice that of the longest side of a similar triangle, say, then the shortest side will also be twice that of the shortest side of the other triangle, and the median side will be twice that of the other triangle. Also, the ratio of the longest side to the shortest in the first triangle will be the same as the ratio of the longest side to the shortest in the other triangle. The crucial fact is that two triangles are similar if and only if their corresponding angles are equal, and this occurs for example when two triangles share an angle and the sides opposite to that angle are parallel.
Using right triangles and the concept of similarity, the trigonometric functions sine and cosine can be defined. These are functions of an angle which are investigated in trigonometry.
In the sequel, we will consider a triangle with vertices A, B and C, angles α, β and γ and sides a, b and c. The side a is opposite to the vertex A and angle α and analogously for the other sides.
The sum of the angles α + β + γ is equal to two right angles (180 degrees or π radians). This allows to determine the third angle of any triangle as soon as two angles are known.
A central theorem is the Pythagorean theorem stating that in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. If γ is the right angle, we can write this as
c2 = a2 + b2
This means that knowing the lengths of two sides of a right triangle is enough to calculate the length of the third — something unique to right triangles. The Pythagorean theorem can be generalized to the law of cosines:
c2 = a2 + b2 – 2abcos(γ)
which is valid for all triangles, even if γ is not a right angle. The law of cosines can be used to compute the side lengths and angles of a triangle as soon as all three sides or two sides and an enclosed angle are known.
The law of sines states sin(α) / a = sin(β) / b = sin(γ) / c which can be used to compute the side lengths for a triangle as soon as two angles and one side are known. If two sides and an unenclosed angle is known, the law of sines may also be used; however, in this case there may be zero, one or two solutions.
Points, lines and circles associated with a triangle
A perpendicular bisector of a triangle is a straight line passing through the midpoint of a side and being perpendicular to it, i.e. forming a right angle with it. The three perpendicular bisectors meet in a single point, the triangle’s circumcenter; this point is the center of the circumcircle, the circle passing through all three vertices. The diameter of this circle is given by a/sin(α).
Thales’ theorem states that if the circumcenter is located on one side of the triangle, then the opposite angle is a right one. More is true: if the circumcenter is located inside the triangle, then the triangle is acute; if the circumcenter is located outside the triangle, then the triangle is obtuse.
An altitude of a triangle is a straight line through a vertex and perpendicular to (i.e. forming a right angle with) the opposite side. This opposite side is called the base of the altitude, and the point where the altitude intersects the base (or its extension) is called the foot of the altitude. The length of the altitude is the distance between the base and the vertex. The three altitudes intersect in a single point, called the orthocenter of the triangle. The orthocenter lies inside the triangle if and only if the triangle is not obtuse. The three vertices together with the orthocenter are said to form an orthocentric system.
An angle bisector of a triangle is a straight line through a vertex which cuts the corresponding angle in half. The three angle bisectors intersect in a single point; this point is the center of the triangle’s incircle, the circle which lies inside the triangle and touches all three sides. There are three other important circles, the excircles; they lie outside the triangle and touch one side as well as the extensions of the other two. The centers of the in- and excircles form an orthocentric system.
A median of a triangle is a straight line through a vertex and the midpoint of the opposite side. The three medians intersect in a single point, the triangle’s centroid. This is also the triangle’s center of gravity: if the triangle were made out of wood, say, you could balance it on its centroid, or on any line through the centroid. The centroid cuts every median in the ratio 2:1, i.e. the distance between a vertex and the centroid is twice as large as the distance between the centroid and the midpoint of the opposite side.
The midpoints of the three sides and the feet of the three altitudes all lie on a single circle, the triangle’s nine point circle. Its radius is half that of the circumcircle. It touches the incircle and the three excircles.
The centroid, orthocenter, circumcenter and center of the nine point circle (but not necessarily the center of the incircle) all lie on a single line, known as Euler’s line. The center of the nine point circle lies at the midpoint between the orthocenter and the circumcenter, and the distance between the centroid and the circumcenter is half that between the centroid and the orthocenter.
If one reflects a median at the angle bisector that passes through the same vertex, one obtains a symmedian. The three symmedians intersect in a single point, the symmedian point of the triangle.
Computing the area of a triangle
The area S of a triangle can be computed in several ways. The most commonly used formula is:
S = 1/2 × base × altitude
where the altitute can be chosen arbitrarily. This formula shows that in the figure
the two triangles ABC1 and ABC2 have the same area, since the lines AB and C1C2 are parallel.
Another way to compute S is Heron’s formula:
where s = 1/2 (a + b + c) is one half of the triangle’s perimeter.
Alternatively S = sr where s is defined as above and r is the radius of the triangle’s incircle,
where AB and AC are the vectors pointing from A to B respectively C, and |AB × AC| denotes the length of their cross product. This is because |AB × AC| represents the area of the parallelogram formed by these vectors, and thus the area of the triangle is half this.
If the vertex A is located at the origin (0,0) of a Cartesian coordinate system and the coordinates of the other two vertices are given by B = (x1, y1) and C = (x2, y2), then the area S can be computed as 1/2 times the absolute value of the determinant
i.e.
Angles
Angles are basic to our observations of the world. We think about angles all the time, in our perception of reality, even though we my not be aware of it at any given instant (an in fact most of us think about angles without ever being aware of it). Since angles are measurable quantities we have to have a way of specifying how to measure angles and what numbers and units to give them.
One basic way of measuring angles is to start somewhere on the plane geometrical surface – a good, flat landscape – and say that if we go around one revolution, completing a circle, we have covered 360 degrees. Thus, the unit of measurement of angles will then be degrees and 360 degrees = one revolution. The degree symbol is usually the superscript o.
A portion of a revolution will of course be smaller than 360 degrees. For instance, if we go round 1/4th of a circle, as shown in bottom part of the figure below., then we cover only 1/4th of 360 degrees, which is equal to 90o.
Area
Since the triangle is a bounded figure one can define an area that it encloses. This might seem complicated to do, if you’re just staring at a random picture of a triangle. But, one can look at a very simple triangle first and see how we can find out its area. The result is a simple relationship that actually generalizes to all triangles if one correctly identifies the base and the height. The way this is done is as follows: one simply picks a side of the triangle and calls it the ‘base’ and then from the edge opposite to that side one drops a perpendicular line to the side that is picked as ‘base’ … this perpendicular line is the height. The following figure may help you imagine how this is done. The area of the triangle is then one-half the base times the height:
Notice that the base and the height both have units of length and so the area A will have units of length2, as expected.
Derivation of Area of a Triangle
The simplest way to try to arrive at the area of a triangle is to look at our simple – “neat” – right triangle. Recall from the section on geometry, a kind of figure that we called a rectangle. Basically, the rectangle is what we defined as a closed figure with unequal length and width as its sides. The area of a rectangle is simply length x width. Now, imagine drawing a line from one corner of the rectangle to the opposite corner. This line is called the diagonal line and – as you can see in the figure below- it actually splits the rectangle in half. Each half of the rectangle is a right triangle because the adjacent sides of the rectangle are perpendicular to each other.
One of the right triangle regions is lined with a grid. This gridded region occupies exactly half of the area enclosed by the rectangle. Thus, the area of this gridded region – which is the area of a right triangle of one leg of length L and another leg of length W – is equal to half the area of the rectangle: Area gridded region = 1/2 L x W.
For the triangle we can consider the length L to be equivalent to the base and the width W to be equivalent to the height. Thus, the area of the right triangle is: Area = 1/2 (base) x (height).
Circles The researcher consider the basic characteristics of a circle since the inscribe circle was related to the area and perimeter of a triangle.
The circle is a closed geometric figure as shown in the following figure:
It is defined such that all the points on the circle are at a constant distance from a center. This distance is called the radius of the circle, as indicated in the diagram above.
Circumference
The quantity we call the “circumference” of a geometrical figure is basically the distance one would travel in going around a closed shaped figure once. One can define a circumference for any closed geometric figure. The circumference of a circle is a rather useful quantity, and it depends very simply on the radius of the circle:
C = 2 × p × r
where: C = Circumference, r = radius, and p= 3.1415 ….
You can think of this quantity as a constant that allows you to define the circumference of a circle. The value of p also allows us to define other ways of measuring angles (in units called radians.)
Area
Again, as for any closed geometric shape, we can also define an area that is enclosed by a circle. This is called the area of the circle and it also depends on the radius of the circle. It is given by:
A = p × r2
Notice that the value of p comes in again into this formula (as it does for any geometrical measurement of a circle). Also important is to compare this with the circumference of a circle. While the circumference of a circle depends on r its area depends on r2.
This observation is significant in terms of units. The circumference will always have the units of distance, while the area will always of units of distance2. You can read more about this in the section on powers.
Credit:ivythesis.typepad.com
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