* MÔI TRƯỜNG GSP  Geometer Sketchpad
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Geometer's Sketchpad in Mathematics
Last Updated: February 27, 2003
 It demonstrates how the area formula of a circle can be derived by dividing a circle into small sectors.
 File size: 2KB
 Instructions for use:
 Drag the point labelled "Drag it" to turn the circle into n sectors arranged in a straight line, or vice versa.
 Click the "Show Conclusion" button to see how the area formula can be derived from the diagram.
 It demonstrates how the curves r = a(1 + cos), r = acos2 and r = acos3are formed as increases from 0 to 2.
 File size: 6KB
 Instructions for use:
 The zip file consists of three sketchpad files: cardioid.gsp, 2rose.gsp and 3rose.gsp.
 Open any one of these three files and drag the point on the circle to see the locus of P(r, ) as varies.
Ellipse as a locus II Contribution from Miss Carol Chan
 It demonstrates how an ellipse can be constructed from a variable point on a circle.
 File size: 1KB
 Instructions for use:
 The teacher should first of all explain how the diagram is constructed: A and C are two fixed points and D is a variable point on a circle whose centre is A. EF is the perpendicular bisector of CD.
 Click the "Animate" button to show the locus of F as D varies, and ask the students to notice the relation between the lengths of CF and AF.
 Ask the students why the sum CF + AF remains constant, and what this constant is.
 It demonstrates the geometric meanings of addition, subtraction, multiplication and division of complex numbers.
 File size: 18KB
 Instructions for use:
 The zip file consists of four sketchpad files: Addition.gsp, Subtraction.gsp, Multiply.gsp and Division.gsp
 Open any one of these four files and drag the points marked z_{1} and z_{2} to see how their sum, difference, product and quotient are affected.
 It shows how the slope of a tangent can be approximated by a "chord" of a curve.
 File size: 2KB
 Instructions for use:
 Doubleclick the "Show the tangent" button to show the tangent at P.
 Click at any empty space and then select the tangent again.
 From the "Measure" menu, choose "Slope". The slope of the tangent appears.
 Doubleclick the "Show the chord" button to show the point Q and the chord PQ.
 Click at any empty space and select the blue line PQ again.
 From the "Measure" menu, choose "Slope". The slope of the PQ appears.
 Drag the point Q towards P and observe how the slope of PQ approaches that of the tangent.
 Illustration of the theorem "Tangent Perpendicular to Radius".
 File size: 6KB
 Instructions for use:
 Teachers should ask students for the value of OTS before doubleclicking the "Show the angle" button.
 Drag the point S towards T and observe how OTS increases while TOS decreases. The constant relation between two angles is emphasized by doubleclicking the "Show the relation" button.
 Drag the point S further until it overlaps with T (PQ is then a tangent to the circle). TOS should then be 0^{o} while OTS becomes a right angle.
 It demonstrates how a ellipse can be generated as a locus of a variable point whose distances from two fixed points add up to a constant (in this case, 6).
 File size: 2KB
 Instructions for use:
 Doubleclick the "Animate" button and the point P will move in such a way that the sum of the distances PO and PA remains to be the same.
 It demonstrates how a parabola can be generated as a locus of a variable point which is equidistant from a fixed line and a fixed point.
 File size: 2KB
 Instructions for use:
 Doubleclick the "Show the locus" button and the point P will move in such a way that the distance PA remains to be the same as the distance d.
 Illustration of the graphical method in Linear Programming
 File size: 3KB
 Instructions for use:
 The underlying problem is to find the maximum value of 5x + 3y subject to the constraints x 0, y 0, x + y 8 and 2x + y 10.
 Drag the point on the horizontal yellow line to see how the equation of the red line changes with its position.
 Illustration of the locus in Exercise 11C Question 13 (Canotta Additional Mathematics)
 File size: 4KB
 Instructions for use:
 Drag the point marked 't' along the horizontal line to see how the straight lines L_{1} and L_{2} varies with different values of t. The intersection point P then traces out a locus.
 Doubleclick the "Show the eqt. of line" button to show the equations of the straight lines L_{1} and L_{2}. You can use them to solve for the coordinates of P.
 Doubleclick the "Show the answer" button to show the coordinates of P. It should then be obvious that the locus of P is y = x.
 Solution of Supplementary Exercise 3 Question 32 (Canotta Mathematics)
 File size: 7 KB
 Instructions for use:
 Doubleclick the "Show 1st solution" button to show the lines that should be constructed.
 Doubleclick the "Show the answer" button to display the calculation.
 Doubleclick the "Show 2nd solution" button to show the coordinates of the points on the xaxis.
 Doubleclick the second "Show the answer" button to see how the radius can be calculated from the coordinates in step 3.
 Illustrates the centerradius form of circles
 File size: 6 KB
 Instructions for use:
 Doubleclick the "Show Theory" button to show how the equation of circles are derived..
 Doubleclick the first "Hide" button and then display a simple example by doubleclicking the "Show Examples" button.
 Drag the point D to different positions to see how the equation of the circle changes with the radius.
 Hold down the Shift key to select both points C and D, then drag the point C around to see how the equation of the circle changes with the coordinates of the circle.
 Explains the concept of Family of straight lines
 File size: 3KB
 Instructions for use:
 Doubleclick the "x + y  2 + 2(2x  y + 1) = 0" button to show the third straight line. Notice that it also passes through P. Can you explain why it is so?
 Doubleclick the first "Hide" button and then display another straight line by doubleclicking the
"x + y  2 + 4(2x  y + 1) = 0" button. What do you observe?  Doubleclick the second "Hide" button and doubleclick the "x + y  2 + k(2x  y + 1) = 0" button to show a thick red line together with the corresponding value of k.
 Doubleclick the "Change the value of k" button to see how the position of the red line changes with the value of k. You should observe that with different values of k, the equation x + y  2 + k(2x  y + 1) = 0 represents a family of straight lines which have different slopes and all pass through P.
 Suppose an aeroplane is flying at a constant height above your head, will its angle of elevation (as observed from you) remain unchanged as well? Have a look at this vivid presentation!
 File size: 7KB
 Instructions for use:
 Doubleclick the "Display eye level" button to see the horizontal level of the observer's eyes.
 Doubleclick the "Display angle of elevation" and "Display planar diagram" buttons to mark and measure the angle of elevation.
 Drag the point B or doubleclick the "Catwalk" button to see how the angle of elevation changes while B moves.
 When is the angle of elevation largest? Doubleclick the "Show closest point" button and repeat step 3!
 This is the Part II of Another version of Angles in the Same Segment: actually the initial diagram is just the same; but what about if the chords CD and EF do not have an intersection point inside the circle?
 File size: 6KB
 Instructions for use:
 Notice that GC x GD = GE x GF where G is the intersection point of CD and EF.
 Drag the point D towards the point U. Is it still meaningful to talk about the equality in step 1 when CD and EF do not intersect inside the circle?
 Can you explain the relation by making use of "Exterior Angle of Cyclic Quad."? Doubleclick the "Explain" button.
 A short question on Tangents to a Circle together with stepbystep solution.
 File size: 25KB
 Instructions for use:
 Doubleclick the "Show" button to see the first part of the solution and bring on the next "Show" button.
 Doubleclick the second "Show" button to see the second part of the solution and bring on the third "Show" button, and so on.
 It demonstrates a result which is equivalent to "Angles in the Same Segment", but expressed in terms of lengths rather than angles. This file also has a PART II coming soon!
 File size: 5KB
 Instructions for use:
 Look at the lengths of GC, GD, GE, GF. Can you guess any relation between them?
 Doubleclick the "Show their relation" button.
 Drag the various points to see whether the equality relation remains true. (Of course CD and EF must have an intersection point)
 Can you explain the relation by making use of "Angles in the Same Segment"? Doubleclick the "Explain" button.
Circle inscribed in a quadrilateral Contribution from Mr. Grandy Lit
 It demonstrates how the derivative can be used to find the maximum or minimum points of a graph.
 File size: 3KB
 Instructions for use:
 For the best effect, choose "Preferences" from the "Display" menu. In the "Slope and Calculation Precision" dropdown list, choose "tenths" and then press "OK"
 Doubleclick the "Show the slope of tangent" button
 Drag the point P to the right hand side. What is the slope of the tangent at the maximum point (3, 0)? And how does the sign of the slope change when P passes through (3, 0) from left to right?
 Drag the point P further to the right hand side. What is the slope of the tangent at the minimum point (3, 4)? And how does the sign of the slope change when P passes through (3, 4) from left to right?
 If a circle is inscribed in a quadrilateral whose three of the sides are known, how can the remaining side be determined?
 File size: 3KB
 Instructions for use:
 Doubleclick the "Show" button to see the method and the answer at the bottom
 Doubleclick the "Hide" button to hide the solution
 Drag the points A, B, C or D to a slightly different position to generate a new problem
 This time, work out the answer by yourself and then check it by doubleclicking the "Show" button
 It demonstrates the theorem about opposite angles in a cyclic quadrilateral
 File size: 5KB
 Instructions for use:
 Doubleclick the "Show the sum" button to get the sum of the angles P and R.
 Doubleclick the "Move the point Q" button or drag the point Q yourself to see how the sum remains constant while the two angles change.
 Click your mouse button to stop the animation. Then try the "Move the point S" button.
 Click the "Show me why it is so" button to see a brief explanation of the theorem.
 It shows how the coefficients of a quadratic function affect the appearance of its graph
 File size: 5KB
 Instructions for use:
 Drag the point marked with "a" from the positive side to the negative side of the real number line. What effect does it have on the graph? In particular, what happens to the graph when a = 0?
 Drag the point "a" to somewhere near 0.5. Then drag the point "b" in both directions along the real number line. What effect does it have on the position of the vertex?
 Drag the point "a" to somewhere near 0.5. Then drag the point "b" again. How is the effect different from that in step 2?
 Drag the point "c" in both directions along the real number line. Does the shape of the graph change? Or does the position of the graph change?
 It shows how a circle can be described by the parametric equations x = cos t, y = sin t.
 File size: 3KB
 Instructions for use:
 Doubleclick the "Hide" button in the diagram to hide some of the words and lines.
 Doubleclick the "Animate" button and see how the point P moves when t increases.
 Click your mouse button again to stop the animation.
 It shows how an ellipse can be described by the parametric equations x = 2cos t, y = sin t
 File size: 2KB
 Instructions for use:
 Doubleclick the "Animate" button and see how the point P moves when t increases.
 Click your mouse button again to stop the animation.
 It shows how the vertical (or horizontal) component of a circular motion gives rise to a simple harmonic motion.
 It can also be used together with "Parametric Equations (1)" to explain why the parametric equations work.
 File size: 2KB
 Instructions for use:
Just doubleclick the "Animate" button and see how the points outside the circle oscillate.
 It shows how an image is formed in a convex mirror and how its position varies with the object or the focus.
 It also demonstrates the formula 1/u + 1/v = 1/f, where u, v,, f stand for the distance between the object and the mirror, the image and the mirror, the focus and the mirror respectively.
 File size: 3KB
 Instructions for use:
 Doubleclick the top "Show" button to show the image.
 The object and the focus can now be dragged to show how they affect the position and the size of the image.
 Doubleclick the green "Show" button to show the various distances.
 Note that 1/u + 1/v and 1/f are equal. (In convex mirror, v and f are taken to be negative).
 The object and the focus can now be dragged to show that the relation mentioned in 4 is always true.