A selection of Geometry constructions using ClassPad

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1 A selection of Geometry constructions using ClassPad Geoff Phillips Please note: These are unedited, very brief outlines only, but should contain enough detail to provide starting points for geometry investigations using the ClassPad CAS calculator s Geometry application. Rough construction notes are provided at the end of this document, after the diagrams. Names of points may not always seem ideal, as I was learning my way around the Geometry application during construction of these files, and often had to backtrack and add objects out of order. Please refer to the ClassPad manual for detailed instructions. After sucessfully completing constructions, I suggest dragging points to form new constuctions and observe the effect of key objects, measurements and calculations. The activities outlined below will hopefully encourage investigation and use of geometry terms.

2 Aubel Centroid Chordtan Quad tri Traptri Unitcir

3 Anglelin Cevians circgeom excircle incentre midquad

4 semicirc theorem sinerat parabcon Thebault s theorem Ptolemy s theorem Inscribed square

5 Incentre ratios Circle square circle triangle squares squares 4 lunes Quadr rectangles Intersecting circles

6 Quad parallelogram Tri square rectangles Triangle area animation sine rule Gradient chord geom 2

7 Aubel Start with any quadrilateral. Construct a square on each side and find the centre of each square using the intersection of diagonals. Join the centres of opposite squares with segments. What do you notice about the two segments? Centroid Start with any triangle. Join each vertex with the midpoint of the opposite side. Do these segments always intersect at the same point? (This point is called the centroid.) Measure the distances from centroid to corner and centroid to midpoint. Calculate the ratio of these distances. Is it always the same? Chordtan Draw a circle and construct a tangent to it at C. From a point D on the tangent, draw a segment DE that cuts through the circle at F and G. Measure CD, DG and DF. Calculate CD^2 and DG DF and compare for various circles. Quad tri Construct a quadrilateral ABCD. Draw a line on AB. Construct another line CE parallel to BD. Find the areas of triangle ADE and quadrilateral ABCD and compare for various shapes. Traptri Draw a trapezium ADEC. Construct a line through E parallel to side AC. Construct diagonal CD. Find the intersection F, of CD and EB. Shade triangles ABF and DFE and find and compare their areas. Unitcir See diagram. Note that sine is length KG and cosine is centre to K. Anglelin Investigate the angles on a straight line. Cevians A cevian is a point on the side of a shape. Construct a triangle ADE and place cevians (B and C) on 2 sides. Construct BD and CE. Find the mid points M and N of BD and CE. Construct triangle MNE. Compare the areas of the triangles ADE and MNE. circgeom Construct a circle with centre O. Let A, B and C be points on the circumference. Measure and compare angles BCA and BOA.

8 excircle Draw a triangle ABC using three lines. Bisect the angle formed by the lines through AB and BC. Bisect the angle formed by the lines through AB and AC. Mark the point of intersection D of the two previous external angle bisectors. Draw a line through D perpendicular to AB. Mark the point E where the previous perpendicular line intersects AB. Draw a circle centred on D with radius DE. This is the excircle. On the diagram, A refers to the area of triangle ABC, and a refers to the side opposite vertex A etc. Make measurements and calculations as required. incentre Draw a triangle ABC. Bisect each angle and find the point of intersection D of the angle bisectors. Construct a perpendicular from D to side AB, and find the point of intersection E on AB. Draw a circle centred on D with radius DE. This is the incircle. On the diagram, A refers to the area of triangle ABC, and a refers to the side opposite vertex A etc. Make measurements and calculations as required. midquad Draw any quadrilateral. Find the mid point of each side and join these to form an inner quadrilateral. What do you notice? semicirc theorem Construct a semi circle on a base AB (centre O) with point Q on the arc. Draw a perpendicular from Q to AB. Find the point of intersection P of the perpendicular with AB. Make measurements and calculations as shown on the diagram. sinerat Draw a right angled triangle. Make measurements and calculations as shown. Change the triangle and compare the calculated ratio O/H with sin(a). parabcon Draw a line segment AB. Draw a point C and constrain it to fixed co ordinates. Draw a segment ED where D is on AB. Constrain ED to be perpendicular to AB. Select point E only and then Edit/Animate/Trace. Select point D and segment AB only and then Edit/Animate/Add Animation. Select Edit/Animate/Go Once. Thebault s theorem Draw a parallelogram. Draw a square on each side of the parallelogram. Find the centre of each square. Join the centres to form a new quadrilateral. What shape do you get? Alter the original parallelogram and see if the final shape is the same.

9 Ptolemy s theorem Draw a cyclic quadrilateral ABCD. Construct diagonals AC and DB, Measure AB, CD, BC and AD. Calculate AC BD. Calculate AB CD + BC AD. Compare the two calculations. Change the quadrilateral and again compare the calculations. Inscribed square Draw a square CDC D. Draw a point E above the square. Construct segments EA and EB. Construct EF perpendicular to AB. Measure and calculate as shown on the diagram. Incentre ratios Draw a triangle. Bisect each angle to find the incentre D. Measure lengths and perform calculations as shown. Circle square circle. Construct a square. Construct a circle of maximum size inside the square. Circumscribe the square with an outer circle. Measure the area of each circle. Calculate the ratio of the circle areas. triangle squares squares Draw a triangle. Construct a square on each side. Now construct another layer of squares as shown. Find the area of each inner square and each outer square. Find the total area of the inner squares and the total area of the outer squares. How do the total areas compare? 4 lunes A lune is a crescent moon shape. Start with a square and construct a circle that circumscribes it. Construct semicircles on each side of the square. Measure and calculate to find the area of the 4 lunes and compare with the area of the square. Quadr rectangles Construct a cyclic quadrilateral ABCD. Draw a rectangle on side AB that has dimensions AB and CD. (Use a congruency constraint.) Draw a rectangle on side BC that has dimensions BC and AD. (Use a congruency constraint.) Draw a rectangle on side CD that has dimensions CD and AB. (Use a congruency constraint.) Draw a rectangle on side DA that has dimensions DA and CB. (Use a congruency constraint.) Find the area of these rectangles and calculate the total area of the rectangles. Construct and measure diagonals AC and BD. Calculate AC^2 and BC^2 and add these results. Compare this with the total rectangle area.

10 Intersecting circles Draw two intersecting circles. Construct the points of intersection. Draw lines HL and JG as shown on the diagram (H, L, J and G are on the circles and pass through a point of intersection of the circles). Construct HG and LJ. Measure and compare the slopes of HG and LJ. Quad parallelogram Draw any quadrilateral ABCD. Construct diagonals AC and BD. Construct mid points G and H of the diagonals. Join and shade the quadrilateral FHEG. What type of quadrilateral is it? Is this always the case? Tri square rectangles Draw a triangle. Draw perpendicular altitudes on each side. Shade adjacent rectangles as shown. Measure and compare the area of the shaded rectangles. Triangle area animation Construct a segment AB. Translate AB vertically say 4 units to form a parallel segment A B. Construct a point C on A B. Construct triangle ABC. Shade the triangle. Calculate the area of the triangle. Animate C on A B and note the area value. sine rule This is an eactivity that uses a Solve strip. Gradient chord geom 2 This is an eactivity that utilises A Geometry strip and geometry Link as well as calculation lines as shown.

DEFINITIONS. Perpendicular Two lines are called perpendicular if they form a right angle.

DEFINITIONS. Perpendicular Two lines are called perpendicular if they form a right angle. DEFINITIONS Degree A degree is the 1 th part of a straight angle. 180 Right Angle A 90 angle is called a right angle. Perpendicular Two lines are called perpendicular if they form a right angle. Congruent