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Relationships Between Geometry & Algebra

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

In this lesson, we will explore relationships between geometry and algebra. We'll look at some simple and straightforward relationships in these two subject areas as well as one relationship that is less obvious.

Geometry and Algebra

Suppose that during Sally's first year of college, she decides to take algebra and geometry. Though these are both considered to be mathematics courses, the course catalog states that they encompass different subjects within the field.

That is, algebra is described as an area in mathematics that uses variables, in the forms of letters and symbols, to act as numbers or quantities in equations and formulas. Geometry, on the other hand, is described as an area in mathematics that studies points, lines, varied-dimensional objects and shapes, surfaces, and solids.

Initially, Sally expects two very different classes that really don't have too much to do with one another. However, later, she discovers that algebra and geometry are intricately related through certain relationships between concepts in both areas of study. Let's take a look at some of these relationships!

Relationships Between Geometry and Algebra

As we said, algebra has to do with equations and formulas, and geometry has to do with objects and shapes, so how can these two things be related? Well, as one example, you are probably familiar with the fact that an equation can be graphed. For instance, the equation y = x + 3 is the graph of a set of points that satisfy the equation, and it turns out to be a straight line.


alggeomrel2


Already, we see a relationship between algebra and geometry. We can take an equation, which is an algebraic concept, and graph it, making it a geometric concept.

In general, the equation of a line has the form y = mx + b, where m is equal to the slope of the line (where the slope is the change in y-values divided by the change in x-values from one point to the next on the line) and b is equal to the y-intercept of the line. We see that the variables within the equation (both algebraic concepts) can actually be used to refer to geometric concepts (slope and y-intercept) of the line.


alggeomrel7


Oh my goodness! So many connections to be found between algebra and geometry! Let's see what else we can come up with.

Consider the shapes of a circle, rectangle, and square. These shapes fall under the category of a geometric concept. However, what about the areas of these shapes? To find the areas of each of these shapes, we use algebraic formulas.


alggeom2


Ah-ha! Another relationship between algebra and geometry!

Here's another one! Have you ever heard of the Pythagorean Theorem? It's a theorem that states that if a right triangle has legs of lengths a and b, and a hypotenuse of length c, then a2 + b2 = c2.


alggeomrel8


The theorem itself shows a relationship between geometry and algebra by relating the lengths of the sides of a right triangle (a geometric concept) to an equation (an algebraic concept).

Wow! It's really becoming clear just how much algebra and geometry are related to one another! Most of these examples that we've looked at are pretty simple and straightforward. Let's take a look at something that may be a little less obvious.

Transformations and Functions

There are two concepts, one in algebra, and one in geometry, that you may be familiar with, but may not have noticed a connection between. Those are transformations (a geometric concept) and functions (an algebraic concept). You see, a transformation is something we apply to a two-dimensional shape to change its position or size. The four main types of transformations are as follows:

  1. Reflections - flipping an object over a line
  2. Translations - sliding an object
  3. Rotations - turning an object around a point
  4. Resizing - making an object bigger or smaller

For example, sliding a square 4 units to the right is a translation.

Now, consider functions. A function can be thought of as a machine. It takes an input, performs some function on the input, and spits out an output, where any input corresponds to exactly one output. Algebraically speaking, many equations are functions. For example, the equation y = 2x takes an input x, multiplies it by 2 and gives the output y.

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