Area Tutorial 3 – Area of a Parallelogram

In the previous area computation tutorials, we have learned how to compute the area of a rectangle and the area of a triangle.  In this tutorial, we are going to learn how to compute the area of a parallelogram.

In Figure 1, we have parallelogram ABCD with given base and the dashed segment as its height. If we drop down a vertical segment from point C and extend a horizontal segment from D to the right, we can form triangle CDF as shown in Figure 2.

Figure 1 – Parallelogram ABCD with a given base and height.

Now, angle ABE is congruent to angle DCF (Why?), AB is congruent to CD, and angle BAE is congruent to angle CDF. Hence, by ASA congruence postulate, triangle ABE is congruent to triangle DCF.

Figure 2 - ...

Since triangle BAE is congruent to triangle CDF, we can move ABE to coincide with DCF forming the rectangle in Figure 3. Click here to explore the translation using GeoGebra.

Figure 3 – Triangle ABE is translated and is superimposed to triangle CDF.

Since BCFE is a rectangle, its area therefore is the product of its base (length) and its height (width). We removed nothing from the parallelogram, therefore, the area of the parallelogram is the same as that of the area of the rectangle. Thus, the area of a parallelogram is the product of its base and its height.

Mr. Pilarski has almost a similar explanation but in video format.

Enjoy and Learn More

Introduction to the Concept of Functions

Problem: A cube is painted on all faces and cut into smaller cubes of the same size. Investigate the number of painted faces of the smaller cubes.

Discussion

In Figure 1, shown are the cubes with side lengths two units and three units. The light blue cube has been painted and cut into eight smaller unit cubes, while the yellow cube has been painted yellow and cut into 27 smaller unit cubes. To avoid confusion, we will simply call the bigger or uncut cubes “cube”and the smaller cubes “unit cubes”.

How many unit cubes can be formed from a painted cube with length 4 units? 5 units? n units?

Figure 1 – Cubes with side lengths 2 and 3 cut into unit cubes.

Before scrolling down, investigate the number of painted faces of each yellow unit cube. In Figure 2, the unit cubes have been drawn from different perspectives to make visualization easier.

How many cubes are there with 3 painted faces? 2 painted faces? 1 painted face? 0 painted face?

Figure 2 – Cube cut into 27 unit cubes shown in different perspectives.

Without drawing, can you determine the number of painted faces of a cube with side length 4 units? How about 5 units? 6 units?

Challenge: Find a formula for the number of unit cubes 3 painted faces, 2 painted faces, 1 painted face and 0 painted face given a side length n units.

To determine how many unit cubes are painted given a particular size, it will help us if we know the properties of a cube. Let us recall that a cube has 8 vertices, 6 faces and 12 edges.

Figure 3 – Parts of a cube.

For the sake of discussion, we will name and color the unit cubes (see Figure 4) and group them depending on their positions – whether they are at the edges, vertices or faces of the cube.

  • Vertex Cubes (Red) – are the unit cubes located at the vertices of the cube. It is evident only 3 of their faces are painted.
  • Edge Cubes (Green) – are the unit cubes located at the 12 edges of the cube. Note that only two of their faces are painted.
  • Wall Cubes (Blue) – are the unit cubes at the faces of the cube. We will call the wall cubes because we will use the word “face” in another context.
  • Core Cubes (No color) – are cubes that are at the core of the cube that was not painted.

Figure 4 – The cube showing number of painted faces depending on their positions.

From Figure 4, it is clear that vertex cubes have 3 painted faces, edge cubes have 2 painted faces, wall cubes have 1 painted face and core cubes have no painted face.

Extending this type of grouping to cubes of larger side lengths, a pattern can be seen as shown in the table below. The calculation for the number of cubes can be generalized.

Figure 5 – Table showing the relationship of the number of painted faces given the cube’s sidel length.

Let us make the following definitions:

a = number of core cubes of an cube with side length n

b = number of wall cubes of a cube with side length n

c = number of edge cubes of a cube with side length n

d = the number vertex cubes with side length n

then

a = (n – 2)3 = n3 – 9n2 + 27n – 27

b = 6(n-2)2 = 6n2 – 24n + 24

c = 12(n-1) = 12n- 12

d = 8

That means that we have created a formula for finding the number of painted faces no matter how large it is. For example, if we want to find the number of wall cubes in from a cut cube with length 100 units, then there are a = 6(100)2 – 24(100) + 24 = 57624 wall cubes.

Question: If m is the volume of the cube before it was cut, then how can you express m in terms of a,  b, c and d?

Let us consider first the equation c = 12(n-1) = 12n – 12. Note that for each side length n, there is a corresponding number of edge cubes denoted by c. And there is only one c. For instance, if n = 3, there is no other value for c except 24. And clearly, this is true for all values of n.

If for each n there is exactly one corresponding c, then we say that c is a function of n.  Hence, we describe what a function means:

A function is a relationship between two sets where for every element of the first set, there is exactly one corresponding element in the second set.

In our case, our first set is the side length of the cube which is n, and our second set is the number of edge cubes that we denoted by c.   In general, if we have set A and set B, for every element in A, there is exactly one corresponding element in B.

The symbol f(x) is read as “f of x” and is usually used as a notation of a function.

We now denote the functions a as f(x), b as g(x), c as h(x) and d as k(x).

We will now graph our functions in the xy plane. The graphs of these functions are shown in Figure 6. The dots represent the ordered pairs and the dashed lines and curves are the lines that we will be formed if we will let x be the set of real numbers instead of integers.

Figure 6 – The points and the trendlines of the graphs of f(x), g(x), h(x) and k(x).

Summary

In this article, we have learned the following:

  • A function is a relationship between varying quantities.
  • A function is a relationship between two sets where for each element in the first set, there is exactly one corresponding element in the second set.
  • A function maybe represented as a table, set of ordered pairs such as (3,8), (4,8), (5,8), equations and graphs.

In the sequel of this article, we will discuss more about the basics of functions and its properties, so keep posted.

Wingeom Tutorial 1 – The Midline Theorem

Introduction

Wingeom is a dynamic geometry software created by Philip Exeter University.  It is capable of 2-dimensional and 3-dimensional geometric drawing and construction.

This is the first tutorial of the Wingeom Tutorial Series.  Most of the construction in this tutorial series will deal with 2-dimensions.

The Wingeom Environment

When you open Wingeom, the window shown below will appear. You have to click the Window menu, then choose the environment that you want to display.  Wingeom can construct figures in Euclidean, hyperoblic and spherical plane.

Figure 1 – The Wingeom window.

It is also capable of constructing Voronoi diagrams and tessellations.

Using Wingeom in Exploring the Midline Theorem

In this construction, we will explore the relationship of a triangle and its midline (or midsegment), the segment connecting the midpoints of its two sides as shown in Figure 2.

Figure 2 – Triangle ABC with midline DE.

In the construction below, we will construct 3 points A, B and C and connect them with the segment tool. After drawing the triangle, we will get the midpoints of AB and AC and explore the length and interior angles of the two triangles formed.

To perform the construction, follow the construction steps below and answer the questions.

Construction Steps

1.)    To open the construction window shown in Figure 2, click the Window menu and then click the 2-dim option.

2.)    Next we will show the Wingeom toolbar. The toolbar displays the tool that we can use to draw and manipulate geometric figures.  To display the toolbar, click the Btns menu and then click Toolbar.

Figure 3 – The Wingeom window and its toolbar.

3.)  The first step in our construction, we will draw the vertices of our triangle. To do this, right click three different locations on the drawing pad. Notice that Wingeom automatically names the points in alphabetical order.

4.)    Next, to construct the sides of the triangle, select the segments option button in the toolbar then drag point A to point B to construct segment AB.

5.)    Using the steps in 4, draw segments AC and BC.

6.)    Next, we will draw the midpoint of AB. To do this, click the Point menu and then click on Segment… to display the new point dialog box.

Figure 4 – The new point dialog box.

7.)    In the relative to segment, type AB, leave the coordinate to 1/2 and then click the mark button. Notice that point now lies on AB. This means that Wingeom should construct a segment halfway of AB.

8.)     To create a midpoint of AC, delete the text in the relative to segment text box and type AC. Then click the mark button.  Notice that point E now lies on AC. Click the close button in the new point dialog box to finish.

9.) Draw segment DE. Refer to step 4.

10.) Let us see what happens if we drag the vertices of the triangle. To drag the vertices of the triangle, click the drag vertices option button on the toolbar, then drag the vertices of the triangle.

11.) Now, we will display the length of DE and BC. To display the length of DE. To do this, click the Meas menu to display the measurements dialog box. Type DE in text box of the measurements dialog box and then click the ENTER key.

Figure 5 – The measurements dialog box.

12.)  Next type BC in the text box and then press the ENTER key. What can you observe about the lengths of segments BC and DE?

13.) Select the drag vertices option button on the toolbar and drag the vertices of the triangle.  Is your observation still the same?

14.) Next, we will try to observe the relationship among the interior angles of two triangles – triangle ABC and triangle ADE. First we will display the measure of angle ABC. To do this, type <ABC in the text box of the measurements dialog box and press the ENTER key.

15.) Display the measures of the following angles using step 14: ADE, AED and ACB.

16.) What do you observe about the measures of the interior angles of triangle ABC?

17.) Close the measurements dialog box and drag the vertices of the triangle (refer to step 13). Are your observations still the same?

18.) Based on your observations, make a conjecture about the relationship of triangle ABC and its midline DE.

19.) Prove your conjectures.

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