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TAITA COLLEGE MATHEMATICS DEPARTMENT YEAR 10 GEOMETRY BOOKLET NAME: _________________ GEOMETER’S TOOLS REFERENCE PAGE Open Geometer’s sketchpad by clicking twice quickly on the icon. The tools for your work are located down the left hand side. Test each of these out. ARROW TOOL: This is used to highlight lines/points/circles etc on the workpage. By using the appropriate tools, draw an interval and a point not on this line. Label the three points that have been created (should be A, B and C). Use the arrow tool to click anywhere else on the screen. Now point to point C and when the arrow becomes horizontal, click (the point should change to black). Drag the mouse to the approximate centre of the interval you have drawn and when the arrow is horizontal hold down the shift key and click with the left mouse button. Note that to highlight more than one object, you will always need to hold down the shift key. POINT TOOL: This will put a point wherever you click on the workpage. CIRCLE TOOL: Click and hold down the ‘circle’ tool and drag it to the size of circle you need. LINE TOOLS: Click on the ‘line’ tool with the left mouse button and keep it held down. A menu will display a ‘line segment’ tool, a ‘ray’ tool and a ‘line’ tool. To draw with any of these you make your first point by clicking and holding down the left mouse button then dragging it to where you want to put your 2 nd point. Practise doing this with the three tools. TEXT TOOL: You can add text to your diagrams if you wish. Note that you do not need to use this to label points and lines. Once you are able to draw with these tools, you can use the various menus at the top of the screen to investigate geometrical properties. THE ANGLE IN A SEMI-CIRCLE INVESTIGATION D Select the segment tool and construct a segment in the middle of the page. Highlight the segment (by using the arrow tool (it will turn pink)) and Construct it’s Midpoint. Now highlight the midpoint and a point at the end of the C A B segment. Construct a Circle by Center + Point. You now have a circle with diameter. Draw a line from one end of the diameter to another point above this line on the circle. Now join this line to the other end of the diameter. Your diagram should look like the one at right. Measure the straight angle by clicking on the three vertices on the diameter and choosing Measure then Angle. Now measure the size of the angle at the circumference. Click on vertex D and drag it around the circle. Notice that the angle stays 90. EXERCISE 1 Calculate the value of x in the following: 1) 3) D D x C C A B x 29 A B x = ________ x = ________ 2) 4) D C B A C x B x D x = ________ x = ________ A TANGENT PERPENDICULAR TO RADIUS This is a difficult property to prove using the sketchpad so we will prove that if we draw a perpendicular line to a radius of a circle, that this line is a tangent (that means that it only touches the circle at one point called the point of contact). Draw a large circle in the middle of the sketchpad. Use the arrow tool to click elsewhere on the screen (to unhighlight the circle), then highlight the two points that have been created and Construct a Segment (the radius of the circle). Keep the highlight on this radius and also highlight the point on the circle and Construct a Perpendicular line. Notice that this line is tangent to the circle. Place a point on the tangent line and highlight the centre of the circle. Construct a Segment. Measure all the angles in the triangle that you have created. Highlight the point of contact and the point on the tangent line. Go to the Display menu and Animate points. Write down what you notice about the measurements for your three angles. __________________________________________________________________________________ __________________________________________________________________________________ Complete: A tangent to a circle meets the radius at ____ at the point of contact. EXERCISE 2 Find the value of x in the following: 1) 2) 3) B 39 32 x x x A 29 C x = ___________ Reason: ______________ ______________________ ______________________ ______________________ ______________________ x = __________ Reason: ______________ ______________________ ______________________ ______________________ ______________________ x = ________ Reason: ______________ ______________________ ______________________ ______________________ ______________________ INTERIOR / EXTERIOR ANGLE SUM OF A QUADRILATERAL Use the ‘ray’ tool to draw a quadrilateral. Measure the interior angles of the quadrilateral and write your results in the space provided: ABC = ___________ CDA = ___________ BCD = ___________ DAB = ___________ Use the Measure menu to Calculate the angle sum of these interior angles. Go to the Measure menu and choose Calculate. Now click on your measurement for ABC then click on + on the calculator then BCD then + then CDA then + then DAB. Now click on OK. Write down the value of your measurement here: Angle sum of the quadrilateral = ___________________ Click on any of the vertices of the quadrilateral and drag to other places on the screen (make sure that you do not cross any of the other sides). What do you notice about the interior angle sum? Write your findings below. __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ Complete: Interior angle sum of a quadrilateral = __________ To calculate the exterior angle sum it is necessary to draw points on each of the sides produced of the quadrilateral. Start on AB produced then BC produced, then CD produced, then DA produced. Label these points by right clicking on the point and choosing Show Label. Your diagram should look similar to the one at right. H Measure the exterior angles of this quadrilateral and write your answers below: HAB = ___________ EBC = ___________ E B C F D A G FCD = ___________ GDA = ___________ Go to the Measure menu and Calculate the angle sum of these exterior angles. Click on any of the vertices of the quadrilateral (A, B, C, D) and drag to other places on the screen (make sure that you do not cross any of the other sides). What do you notice about the exterior angle sum? Write your findings below. _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ Complete: Exterior angle sum of a quadrilateral = ______________ Complete the exercise on the next sheet. EXERCISE 3 Find the value of x in the following (give reasons for your answer): (a) (b) x 53 85 x 118 80 119 88 (c) x = ____________ x = ___________ Reason: ___________________ Reason: ___________________ __________________________ __________________________ __________________________ __________________________ x (d) x 95 81 84 82 93 x = ____________ x = ___________ Reason: ___________________ Reason: ___________________ __________________________ __________________________ __________________________ __________________________ (e) (f) x 100 124 84 123 88 x 96 x = ____________ x = ___________ Reason: ___________________ Reason: ___________________ __________________________ __________________________ __________________________ __________________________ EXTERIOR ANGLE OF A POLYGON INVESTIGATION Select the ‘ray’ tool and draw a hexagon. Right click on each vertex (or point) and choose ‘Show Label’. Repeat for each vertex of the hexagon. In order to measure the exterior angles of this polygon, you will need to create points on the 6 rays. Use the point tool to create these points starting from AB produced then BC produced and so on. Label these exterior points. Your diagram should now look similar to the one below. Measure each exterior angle and write your results below: LAB = ___________ GBC = ___________ H HCD = ___________ IDE = ___________ JEF = ___________ KFA = ___________ C G D I B Use the Measure menu to Calculate the sum of these exterior angles. L A Click on any of the vertices of this hexagon and drag it to a different place on the screen (be careful not to cross other sides). What has occurred to the sum of the angles. Write about your findings below: _________________________________________________ E J F K ____________________________________________________________________ ____________________________________________________________________ Complete: Sum of exterior angles of a hexagon = ________________ Click on each vertex (A to F) in turn moving them closer to the centre of the polygon. Keep moving them until the points are almost overlapping in the middle. This is difficult and takes a bit of time to ensure that lines do not overlap. Has the exterior angle sum changed? Write your findings below: __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ Create other polygons (pentagon, heptagon, octagon, nonagon, decagon) using the sketchpad and investigate the exterior and interior angle sums. Write about your findings in the space below then complete the exercise over the page. EXERCISE 4 Find the value of x in the following (give reasons for your answer): (a) (b) 39º 125º 105 98 xº x 110 86º 146º 37º 75 xº = _____________ xº = _____________ Reason: _____________________ Reason: ______________________ ____________________________ ____________________________ ____________________________ ____________________________ (c) Use your sketchpad diagrams to find the interior angle sum of: (d) Use your sketchpad diagrams to find the exterior angle sum of: Shape Quadrilateral Angle sum Shape Quadrilateral Pentagon Pentagon Hexagon Hexagon Heptagon Heptagon Octagon Octagon Nonagon Nonagon Decagon Decagon Use this space to help you in your calculations Angle sum REFLECTION Use the line segment tool to draw a triangle on the left hand side of the sketchpad. Use the line tool to draw a vertical line to the right of this triangle - this will be your mirror line. While it is still highlighted (coloured pink) click on the Transform menu at the top and choose Mark Mirror. Highlight the triangle (either by drawing a rectangle around it with the arrow tool or by highlighting the three sides individually). Go to the Transform menu and choose Reflect. Your screen should look something like this: Now label the points (or vertices) of the original triangle. Do this by right clicking on each vertex and choosing Show Label. Do the same for the image – write about the labeling system used for the image and it’s relationship to the original triangle. _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ Note that the sense (direction around the object) is reversed under a reflection ie if you travel from point A to point B to point C in a clockwise direction you will need to go in an anticlockwise direction to go from point A to point B to point C. Use the arrow tool to click on either of the points on the mirror line and drag them to other points on the screen. Notice what happens to the reflection. What happens if you drag the mirror line to pass through the bottom right of the original triangle? _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ When a point of the original object is on the mirror line, it is called an invariant point. Draw the following on new sketches, create the mirror line and reflect the objects in the mirror line: Note that the mirror line in each case acts as a line of symmetry. Properties of Reflection Time to do some calculations! Measure the 3 angles of the triangle and measure the 3 angles of its image. What do you notice? ____________________________________________________________________________ Match the pairs of equal angles by drawing a line between them: ABC BAC BAC ABC BCA BCA Test the accuracy of your findings by clicking on any vertex of the triangle and dragging it to other parts of the screen. Make sure that no points are highlighted and highlight A and A. Construct a Segment. Click somewhere else on the screen and highlight B and B. Construct a Segment. Repeat for C and C. Use the point tool to construct points at the intersection of each of these segments and the mirror line – construct the point on AA firstly then BB then CC (you will notice that both lines will turn light blue when you have found the correct spot. Label each of these points in the order that you created them. Use the Measure menu to measure the distances AD and AD. Now measure distances BE, BE and CF, CF. Time for some fun! Right click on point A and choose Animate Point. What do you notice about the distances AD and AD? If you are unsure, slow the speed of the animation down. ____________________________________________________________________________ Whilst the animation is occurring, right click on the other two vertices in turn (B and C) and choose Animate Point. What do you notice about all the distances from the mirror line? _______________________________________________________________________________ _______________________________________________________________________________ To stop the animation click the stop button and it will stop the target point only (change the target point and choose stop again then repeat to stop the whole animation). Complete: Angles are ___________________ under a reflection. The lengths of the sides of an object are ____________________ under a reflection. Points on the mirror line are ____________________ under a reflection. ROTATION Use the line segment tool to construct a quadrilateral to the left of your sketchpad. Click on each of the points of your quadrilateral and go to the Construct menu and choose Quadrilateral Interior. Use the point tool to construct a point to the bottom right of the quadrilateral. Your diagram will look something like this: Click on the individual point, go to the Transform menu and choose Mark Center. Highlight the quadrilateral (by using the arrow tool) and go to the Transform menu again and choose Rotate. A window will appear with a default setting of 90 and the image will be ‘ghosted’ on the screen. Choose Rotate and the transformation will now appear. Notice that the image is rotated in an anticlockwise direction. Repeat this process and choose 180 (make sure that you only have the original quadrilateral highlighted). Repeat again and choose 270. Draw these shapes on the sketchpad and rotate them according to the conditions stated (note that the shift key can make this process easier): a) Rotate 180 about the point A b) Rotate 90 about point A A A Note that by choosing appropriate angles you can create simple designs. Use this method to create the Mitsubishi logo: Use the line segment tool to draw a short line segment diagonally towards the top of the sketchpad like this: Highlight the lower point, go to the Transform menu and choose Mark Center. Now highlight this line segment and Rotate it 240. Construct a point at the end of the rotated segment and make this your new centre of rotation. Highlight the rotated segment and rotate this 300. Construct a point at the end of this rotated segment and make this the new centre of rotation and rotate this segment 240 (which should join up with your first line segment). Highlight the four points that you have created and Construct the Quadrilateral Interior. Your diagram should look as follows: Highlight the bottom point of this shape (a rhombus by the way) and use the Transform menu to make this your point of rotation (Mark Center). Highlight the rhombus and use the Transform menu to Rotate it by 120 about the bottom point. Click elsewhere on the screen then highlight the original rhombus and rotate it by 240 about the bottom point. To tidy your sketch up, you will need to hide the line segments and points on the original rhombus. Do this in turn by right clicking on each point and segment and choosing Hide Point or Hide Segment. You should now have a completed logo. If you wish to change it’s colour, right click on each rhombus and choose the colour option and a colour you want it to be (I think the real one is red). Properties of Rotations: Draw a triangle on your sketchpad. Mark one of the vertices as the centre of rotation. Highlight each vertex and each side and rotate the triangle through 90. Label the points of the original triangle and it’s image. Write down what you notice about the labeling system used. _______________________________________________________________________________ _______________________________________________________________________________ Notice that the sense of the object remains unchanged (or invariant). Measure the angles of the original triangle and it’s image. What do you notice? Prove your assertion by dragging the vertices of the triangle to other places on the screen. Write about your findings below. ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ Measure the lengths of each side of the triangle and its image. Complete the following statements: Angles are ___________________ under a rotation. The lengths of the sides of an object are ____________________ under a rotation. The centre of rotation is the only _____________________ point. TRANSLATION Use the line segment tool to draw a triangle in the middle of your screen. Highlight each vertex and each side of the triangle. Go to the Transform menu and choose Translate. We want to use a Rectangular translation vector (so click on this), and we will use the default setting of 1cm horizontally and vertically. Choose Translate. Draw a horizontal line segment to the left of the sketchpad. Highlight the two endpoints of this segment and Construct a Circle by Center + Point. Right click on the original segment and Hide Segment. Highlight the two points and Measure the Distance between them. Highlight the circle and translate it by the distance just found. To do this choose rectangular again and type your distance into horizontal distance and make the vertical distance 0cm. Then choose Translate. Repeat this process several times so that you end up with circles translated across the page like this: I J This method could be used to draw the Olympic rings (although you will need to do some vertical translation as well). Find the Olympic rings logo on the Net and use it to create an exact replica. Open a new sketch. Go to the Graph menu and choose Grid Form and Rectangular Grid. Draw a point at (-5,0), another at (0,4) and another at (0,2). Use the arrow tool to highlight these three points and Construct the Triangle Interior. Translate this triangle by the vector 6 ie 6 horizontally to the right and 3 vertically down. Write 3 down the co-ordinates of each vertex of the triangle’s image. ___________________________________________________________________ Draw points at (-7,6), (-5,6), (-7, 4) and (-5, 4). Highlight all the points and Construct the Quadrilateral Interior. Right click on this square and change it’s colour to red. Translate the square by the following vectors: a) 2 2 b) 2 2 (c) 2 2 Write down the vector to complete this simple design: Open a new sketch. Go to the Graph menu and choose Grid Form and Rectangular Grid. Use the Graph menu to Plot Points: (-5,5), (-4,5), (-4,4), (-3,4), (-3,3), (-4,3), (-4,2), (-5,2), (-5,3), (-6,3), (6,4) and (-5,4). Construct the Polygon Interior (a cross). Hide the points. Highlight the shape and Translate it by the vector Right click on the translated cross and choose a different colour. 2 . 1 Complete a tessellation involving at least 10 crosses by translating the original shape each time. Your design should only contain two colours and there should be no overlap of colour. If you make a mistake use the Edit menu and Undo. When you have completed your design, you can hide the grid and axes and individual points. Properties of Translations: Open a new sketch. Use the line segment tool to draw a triangle in the middle of your screen. Highlight each vertex and each side of the triangle. Go to the Transform menu and choose Translate. Use any vector you wish (as long as it is rectangular). Label each point of the triangle and it’s image. Measure the angles and sides of the object and it’s image. Write about the sense of the object under a translation and write about invariant properties: ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ Complete the following statements: Angles are ___________________ under a translation. The lengths of the sides of an object are ___________________ under a translation. There are no ___________________ points. ENLARGEMENT Use the line segment tool to draw a small triangle towards the left of your sketchpad. Place a point to the left of this triangle and double click on it (to Mark Center). This is the center of enlargement. Highlight the three sides of the triangle. Go to the Transform menu and choose Dilate (which is the word for enlargement in this program). A small window will appear – use the default setting of Fixed Ratio and notice that this number is a fraction (and you can change the numerator and the denominator). Change the numerator to 2 and the denominator to 1 (you will see an image ‘ghosted’ on the screen). Click on Dilate. Measure the angles of both triangles. What do you notice? __________________________________________________________________________________ Measure the sides of both triangles. Use the calculator to divide the respective sides of both triangles. Compare the figure you obtain each time to the enlargement factor. What do you notice? __________________________________________________________________________________ Highlight the 3 vertices on the original triangle. Construct the Triangle Interior. Do the same to the enlarged triangle. Now, move the vertices of the enlarged triangle so that it is not crossing over the smaller triangle. Measure the area of both triangles. Divide the area of the image by the area of the original object. Compare this with the enlargement factor. What do you notice? __________________________________________________________________________________ Delete the enlarged triangle. Highlight the original triangle and this time Dilate using an enlargement factor of 3 ( 13 ). Measure the area of this triangle and again divide it by the area of the original triangle. What would you expect this result to be if your enlargement factor was 4? 5? __________________________________________________________________________________ You can also construct figures and ‘enlarge’ using fractional enlargement factors. Open a new sketch and draw a quadrilateral on the sketchpad and use the arrow tool to mark one of its vertices as the centre of enlargement. Highlight the quadrilateral and ‘enlarge’ it by a factor of 1 2 . Construct the Quadrilateral Interior of both quadrilaterals and Measure their areas. Divide the area of the image by the area of the original object. What do you notice? __________________________________________________________________________________ __________________________________________________________________________________ Choose words from the right and place them under the appropriate column for an enlargement. Unchanged (Invariant) Changed Sense / Orientation Side Lengths Angles Shape Area
