For the following two applets and problems, you must try to formulate some sort of GENERALIZATION or RULE that governs these VECTOR TRANSLATIONS. That is what is needed to move past the next threshold.

Page 9, Problem 3

Drag the points at the tip or the tail of the green vector until the quadrilateral A'B'C'D' coincides with quadrilateral PQRS. Once the quadrilaterals coincide, drag the vector around by clicking on the middle of the vector. Do you see how the two quadrilaterals are related?

Page 9, Problem 4

Drag the points at the tip or the tail of the green vector to answer the question.

In the applet below, the RED, GREEN, and PURPLE points are the points found in the problem. Drag the center of the circle to (3,0) and drag the radius point of the circle to (-1,0). Next, drag the ORANGE POINT to either (-5,0) or (1,0). Why not do a google search for CIRCULAR INVERSION and see what you find?

Page 9, Problem 9

Click on the check boxes one at a time.

The Tangent Ratio, The Sine Ratio, and the Cosine Ratio

Apply the Pythagorean Theorem to the sides of each triangle, you will find that the triangle have sides that are the same length. By problem 7 on page 9, these two triangles are CONGRUENT. In mathematical terms,

One of the properties of congruent figures is that their corresponding parts are also congruent (we shorten this as CPCTC). Therefore, we have

Page 10, Problem 7

Here are two rhombi that fit the bill (diagonals and sides are not parallel to the rulings on the grid). Of course, these are just two examples. There are infinitely many others. The understanding (moving to the other threshold) comes when you search for patterns in the constructions. The KEY IDEA is that the diagonals of a rhombus are perpendicular bisectors of each other.

Page 10, Problem 8

Page 10, Problem 9

Here is a screen shot of the constructed rectangle. Do you notice anything? What is the significance of it?

## Page 9, Problems 1 and 2

## Table of Contents

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For the following two applets and problems, you must try to formulate some sort of GENERALIZATION or RULE that governs these VECTOR TRANSLATIONS. That is what is needed to move past the next threshold.

Drag the points at the tip or the tail of the green vector until the quadrilateral A'B'C'D' coincides with quadrilateral PQRS. Once the quadrilaterals coincide, drag the vector around by clicking on the middle of the vector. Do you see how the two quadrilaterals are related?Page 9, Problem 3## Page 9, Problem 4

Drag the points at the tip or the tail of the green vector to answer the question.

## Page 9, Problem 6

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In the applet below, the RED, GREEN, and PURPLE points are the points found in the problem. Drag the center of the circle to (3,0) and drag the radius point of the circle to (-1,0). Next, drag the ORANGE POINT to either (-5,0) or (1,0). Why not do a google search for CIRCULAR INVERSION and see what you find?

## Page 9, Problem 9

Click on the check boxes one at a time.

## The Tangent Ratio, The Sine Ratio, and the Cosine Ratio

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## Page 10, Problem 6

Apply the Pythagorean Theorem to the sides of each triangle, you will find that the triangle have sides that are the same length. By problem 7 on page 9, these two triangles are CONGRUENT. In mathematical terms,

One of the properties of congruent figures is that their corresponding parts are also congruent (we shorten this as CPCTC). Therefore, we have

## Page 10, Problem 7

Here are two rhombi that fit the bill (diagonals and sides are not parallel to the rulings on the grid). Of course, these are just two examples. There are infinitely many others. The understanding (moving to the other threshold) comes when you search for patterns in the constructions. The

KEY IDEAis that the diagonals of a rhombus are perpendicular bisectors of each other.## Page 10, Problem 8

## Page 10, Problem 9

Here is a screen shot of the constructed rectangle. Do you notice anything? What is the significance of it?## Page 10, Problem 10

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