Exploring Geometric Relationships Overview

Introduction

When you hear the word "GEOMETRY," probably one of the first people you think about is Pythagoras. Most adults you ask, whether they were good in math or not, remember the infamous Pythagorean Theorem. In this module, we'll explore how Pythagorean's Theorem works and how we can use it to solve for missing lengths of right triangles. While this is the most common way that we think of using Pythagorean's Theorem, we can also use it to find the distance between two points. As we study this fundamental geometric theorem, we'll also take a look at its proof. In other words, we'll see why it always works! Finally, we'll wrap up this module by learning the formulas for finding the volume of the three-dimensional shapes of cones, cylinders, and spheres. We'll use practical applications to interact with the idea of what volume really means and how it changes when looking at, not only three-dimensional shapes, but three-dimensional shapes with curvature. It's going to be a ball!

Essential Questions

How do we use the Pythagorean Theorem to solve for unknown side lengths of triangles?
How do we use the Pythagorean Theorem to find the distance between two points?
What are the formulas for finding the volume of the three-dimensional shapes of cones? cylinders, and spheres and how do we use them to solve real-world problems?

Key Terms

The following key terms will help you understand the content in this module.

Coordinate Point of a Plane - A pair of numbers defining the position of a point on a two-dimensional plane.

Converse of Pythagorean Theorem - If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, the triangle is a right triangle.

Hypotenuse - The hypotenuse is the side of a right triangle that is directly across from the right angle.

Leg of a Triangle - Either of the two shorter sides of a right triangle. These two sides together form the right angle in the right triangle.

Height of Solids - The vertical height (or altitude) which is the perpendicular distance from the top down to the base.

Cone - A three dimensional figure with a circular or elliptical base and one vertex.

Geometric Solid - The collective term of all bounded three dimensional geometric figures.

Rational Number - Any number that can be expressed as the quotient a/b of two integers, with the denominator b not equal to zero.

Perfect Cubes - In arithmetic and algebra, the cube of a number n is its third power. It is the result of the number multiplying by itself three times.

Cubed Root - One of three identical factors of a number that is the product of those factors.

Sphere - The set of all points in space that are equidistant from a fixed point, called the center.

Diameter - The distance across a circle through its center. The line segment that includes the center and whose endpoints lie on the circle.

Right Triangle - A triangle with exactly one right angle.

Distance Formula - An application of the Pythagorean Theorem based on the distance between two points.

Perfect Squares - The product of a rational number multiplied by itself.

Altitude of a Triangle - The perpendicular distance between a vertex of a triangle and the side opposite that vertex. Sometimes called the height of a triangle. Also, sometimes the line segment itself is referred to as the altitude.

Irrational - A real number whose decimal form is non-terminating and non-repeating that cannot be written as the ratio of two integers.

Pythagorean Triples - A set of positive integers, a, b, and c that fit the rule a2+b2=c2.

Literal Equation - An equation with more than one variable. Formulas are literal equations.

Radius - The distance from the center of a circle to any point on the circle. Also, the line segment that has the center of the circle as one endpoint and a point on the circle as the other endpoint.

Coordinate Plane - A two-dimensional surface on which points are plotted and located by their x and y coordinates.

Radical - A symbol that is used to indicate square roots.

Cylinder - A three-dimensional object with two parallel, congruent, and circular bases.

Pythagorean Theorem - A theorem that states that in a right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the legs.

Deductive Reasoning - The process by which one makes conclusions using known facts, definitions, rules, or properties.

Base (of a Polygon) - For two-dimensional figures, any side can be a base. Typically, however, the bottom side, on which the polygon 'sits', is called the base.

Square Root - The square root of a number is a special value that, when multiplied by itself, gives the number.

Volume - The amount of space occupied by an object.

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