February 2010 - Page 4 of 4 - Math and Multimedia

Annotating and Zooming with the ZoomIt Software

February 5, 2010 GB Software Tutorials, Teaching Tools

ZoomIt software enables you to zoom and draw (or annotate) anywhere on your screen. When I said anywhere, I really mean anywhere. You can draw on your desktop, toolbars, widgets or even icons. The examples below show writings on a Powerpoint slide show and two icons.

Figure 1 – ZoomIt is used to annotate the Powerpoint slide while in Slide Show mode.

The ZoomIt drawings, however, are not permanent. Once you press Esc key, all your Zoomit drawings are erased. If you want a permanent writing pad, you can use Miscrosoft PaintBrush or Classic Whiteboard.

Figure 2 – ZoomIt can be used to write even on icons.

Once you install ZoomIt, you just have to click its icon and then press the combination key to activate it. By default, Zoomit allows you to draw by pressing the Ctrl+2 (that is, holding the Control key on your keyboard and then pressing 2) and zoom by pressing Ctrl+1.

Figure 3 – The ZoomIt dialog box.

You can change the combination keys to activate drawing and zooming. You just have to double click the ZoomIt icon to display the dialog box shown in Figure 3. If you want to change the control keys, you just click the text box (for example, Draw w/o Zoom text box), then press the combinations keys that you want on your keyboard.

The Algebraic and Geometric Proofs of Pythagorean Theorem

February 3, 2010 GB High School Geometry, High School Mathematics

The Pythagorean Theorem states that if a right triangle has side lengths $a, b$ and $c$ , where $c$ is the hypotenuse, then the sum of the squares of the two shorter lengths is equal to the square of the length of the hypotenuse.

Figure 1 – A right triangle with side lengths a, b and c.

Putting it in equation form, we have

$a^2 + b^2 = c^2$ .

For example, if a right triangle has side lengths $5$ and $12$ , then the length of its hypotenuse is $13$ , since $c^2 = 5^2 + 12^2 \Rightarrow c = 13$ .

Exercise 1: What is the hypotenuse of the triangle with sides $1$ and $\sqrt{3}$ ?

The converse of the theorem is also true. If the side lengths of the triangle satisfy the equation $a^2 + b^2 = c^2$ , then the triangle is right. For instance, a triangle with side lengths $(3, 4, 5)$ satisfies the equation $3^2 + 4^2 = 5^2$ , therefore, it is a right triangle.

Geometrically, the Pythagorean theorem states that in a right triangle with sides $a, b$ and $c$ where $c$ is the hypotenuse, if three squares are constructed whose one of the sides are the sides of the triangle as shown in Figure 2, then the area of the two smaller squares when added equals the area of the largest square.

Figure 2 – The geometric interpretation of the Pythagorean theorem states that the area of the green square plus the area of the red square is equal to the area of the blue square.

One specific case is shown in Figure 3: the areas of the two smaller squares are $9$ and $16$ square units, and the area of the largest square is $25$ square units.

Exercise 2: Verify that the area of the largest square in Figure 3 is 25 square units by using the unit squares.

Figure 3 – A right triangle with side lengths 3, 4 and 5.

Similarly, triangles with side lengths $(7, 24, 25)$ and $(8, 15, 17)$ are right triangles. If the side lengths of a right triangle are all integers, we call them Pythagorean triples. Hence, $(7, 24, 25)$ and $(8, 15, 17)$ are Pythagorean triples.

Exercise 3: Give other examples of Pythagorean triples.

Exercise 4: Prove that there are infinitely many Pythagorean triples.

Proofs of the Pythagorean Theorem

There are more than 300 proofs of the Pythagorean theorem. More than 70 proofs are shown in tje Cut-The-Knot website. Shown below are two of the proofs. Note that in proving the Pythagorean theorem, we want to show that for any right triangle with hypotenuse $c$ , and sides $a$ , and $b$ , the following relationship holds: $a^2 + b^2 = c^2$ .

Geometric Proof

First, we draw a triangle with side lengths $a, b$ and $c$ as shown in Figure 1. Next, we create 4 triangles identical to it and using the triangles form a square with side lengths $a + b$ as shown in Figure 4-A. Notice that the area of the white square in Figure 4-A is $c^2$ .

Figure 4 – The Geometric proof of the Pythagorean theorem.

Rearranging the triangles, we can also form another square with the same side length as shown in Figure 4-B.This means that the area of the white square in the Figure 4-A is equal to the sum of the areas of the white squares in Figure 4-B (Why?). That is, $c^2 = a^2 + b^2$ which is exactly what we want to show. *And since we can always form a (big) square using four right triangles with any dimension (in higher mathematics, we say that we can choose arbitrary $a$ and $b$ as side lengths of a right triangle), this implies that the equation $a^2 + b^2 = c^2$ stated above is always true regardless of the size of the triangle.

Exercise 5: Prove that the quadrilateral with side length C in Figure 4-A is a square.

Algebraic Proof

In the second proof, we will now look at the yellow triangles instead of the squares. Consider Figure 4-A. We can compute the area of a square with side lengths $a + b$ using two methods: (1) we can square the side lengths and (2) we can add the area of the 4 congruent triangles and then add them to the area of the white square which is $c^2$ . If we let $A$ be the area of the square with side $b + a$ , then calculating we have

Method 1: $A = (b + a)^2 = b^2 + 2ab +a^2$

Method 2: $A = 4(1/2ab) + c^2 = 2ab + c^2$

Methods 1 and 2 calculated the area of the same square, therefore they must be equal. This means that we can equate both expressions. Equating we have,

$b^2 + 2ab + a^2 = 2ab + c^2 \Rightarrow a^2 + b^2 = c^2$

which is exactly what we want to show.

Leave a comment algebraic proofs, geometric proofs, proof of pythagorean theorem, pure mathematics, pythagora's theorem, pythagorean theorem, pythagorean triples, right heron triangles, right triangles

Geonext Tutorial 2 – Constructing a Square

February 1, 2010 GB Geonext, Software Tutorials

In Geonext Tutorial 1, we used the circle and polygon tools to construct an equilateral triangle. In this tutorial, we are going to use the circle, the parallel and the perpendicular line tools to construct square ABEC shown in Figure 1.

Figure 1 - Square formed by construction using circles, parallel and perpendicular line tool.

In constructing our square, we will first construct the a circle with radius AB, construct a line perpendicular to AB, passing through point A (see line CD), construct a line parallel to CD passing through point B (see line EB), then construct a line parallel to AB and passing through C (see line CE). We then hide all the objects except the vertices of the square and line segment AB, and use the segment tool to construct square ABEC. Follow the instructions below to construct the square.

Instructions

	1. Open Geonext. Click the New Board button.
	2. To construct a circle with center A passing through point B, click the Circle tool, click the drawing board to determine the center of the circle, then click another location to determine its radius. Notice that Geonext, automatically names the points in alphabetical order.
	3. Double click the Straight Line icon to display the list of tools. Choose Line segment tool. Observe that the Straight Line tool icon on the toolbar is replaced by the Line Segment icon. Click point A and then click point B to construct radius AB. Figure 2 - Tools that appear when you double click the Straight line tool in the toolbar.
	4. To construct a line perpendicular to AB passing through point A, double click the Line Segment tool to display the related tools. Choose the Perpendicular line tool, click segment AB, then click point A.
	5. Next we construct the intersection of the line constructed in step 4 and the circle. To do this, click the Point tool, then click one of the intersections. Observe that Geonext automatically names the point C and the opposite point D. After step 5, your drawing should look like Figure 3. Figure 3 - The appearnce of the drawing after step 5.
	6. We now construct a line parallel to CD and passing through point B. To do this, double click the Perpendicular line tool to display the list of related tools, and choose Parallel line. Click line CD (not the points), then click point B.
	7. Applying step 6, construct a line parallel to AB and passing through point C.
	8. Using the Point tool, draw the intersection the line parallel to AB, and the line perpendicular to AB passing through point B. Geonext will name this point E.
	9. Next we hide all the objects except the vertices square *ABEC* and segment AB. To hide the objects, double click the Rename tool (located at the bottom of the toolbar), and choose Hide. Click all the objects except points *A, B, C, E* and line AB. Notice that all the clicked objects change colors. These are the hidden objects.
	10. Click the Move tool and move the vertices of the square. What do you observe?
	11. Using the Line segment tool, construct segments AC, CE and BE and we are done.
	12. Move the vertices of the square. What do you observe?
Exercises: Verify that the figure is a square using the angle tool and by revealing the length of the sides (Refer to Tutorial 1). Based on the construction above, prove that *ABEC* is a square.

Leave a comment Constructing a square, dynamic geometry, geonext tutorial, parallel lines lessons, perpendicular lines lessons, Software Tutorials

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