The Area of a Triangle Nipissing University Mathematics Lecture Talks It Is More Complicated Than You Might Think! presents

The area of a geometric plane figure such as a polygon is the measure of the number of square units the object or plane figure is made up of. Area is a physical quantity expressing the size of a part of a surface. Surface area is the summation of the areas of the exposed sides of an object. Square units (e.g. cm 2 ) are used in quantifying the measures areas and surface areas.

A polygon is a closed planar path composed of a finite number of sequential line segments. The straight line segments that make up the polygon are called its sides or edges and the points where the sides meet are the polygon's vertices. If a polygon is simple, then its sides (and vertices) constitute the boundary of a polygonal region, and the term polygon sometimes also describes the interior of the polygonal region (the open area that this path encloses) or the union of both the region and its boundary. A triangle is a polygon with three vertices and three sides which are straight line segments.

A B C The Vertices The sides The area The Triangle

Computing the area of a triangle Using geometry The area S of a triangle is S = ½bh, where b is the length of any side of the triangle (the base) and h (the altitude) is the perpendicular distance between the base and the vertex not on the base. This can be shown with the following geometric construction.

Applications of Mathematics Some Trivial Applications: handle your money tell time gamble construct objects paint things Some Exciting Applications: solve problems create new formulas discover new relationships

Distance Away from Us Speed Away From Us Edwin Hubble’s graph Galaxies In the Universe The Big Bang What is the scientist (Hubble) trying to convey to us here?

What is the artist (Pablo Picasso) trying to convey in these paintings?

What meaning is the mathematician trying to covey from this odd-looking expression? x 1 y 1 x 2 y 2 x 3 y 3 x 1 y 1 Actually it represents a new formula for finding the area of a triangle when its vertices on the Cartesian plane are known! Add the “down products” and subtract the “up products” and take ½ of this result.

x 1 y 1 x 2 y 2 x 3 y 3 x 1 y 1 The formula can be extended to finding the area of any convex polygon on the Cartesian plane: Add the “down products” x n y n  and subtract the “up products”. Remember that the points must be in counterclockwise order.

Congruent triangles are triangles with the same shape and size In geometry, two sets are called congruent if one can be transformed into the other by an isometry, (i.e., a combination of translations, rotations and reflections). In less formal language, two sets are congruent if they have the same shape and size, but are in different positions (for instance one may be rotated, flipped, or simply moved). A C F B D E We write:  ABC   DEF and thus must have the same area.

There are actually four conditions under which triangles can be proven to be congruent. Each requires that three parts of the triangle (angles or sides) be respectively equal in each triangle. RHS (you are given a right angle, the hypotenuse and one other side) SAS (you are given two sides and the contained angle) SSS (you are given three sides) AAS (you are given two angles and any side) There is also an interesting fifth case: you are given two sides and the angle is not contained. Two possible triangles exist here (the ASS case).

b The SAS case (the given angle must be contained between the given two sides): C A B a c b and sides “a” a nd “b”, c a C find side “c”. Given  C here we use the cosine law Now let’s find its area

b The SSS case: C A B a c b Given sides “a”, “b” and “c” c a C find one of the angles (  C in this case). here we use the cosine law Now let’s find its area

b The AAS case (the given side does not have to be contained – just find the 3rd angle by subtracting from 180  ): C A B a c and side “c”, c find side “a”. Given  A and  C a A C here we use the sine law Now let’s find its area

b The ASS case (the given angle is not contained between the given two sides): C A B a c b and sides “a” a nd “b”, a find angle  B. Given  A A B here we use the sine law and two triangles may be found

A C B b aa a B c c Two Triangles b A C a < b We use the sine law for each of the ASS cases.

A C B b aa a B c c One Triangle a > b b C A

One Triangle (special case) when a = b sin A C B A a b c a  B = 90  B Now let’s find their areas

Summary of the area formulas: The RHS case: The SAS case: The SSS case: The AAS case: The ASS case: where

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