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# «Geometry Expressions Manual TM Saltire Software PO Box 230755 Tigard, OR 97281 0755 © 2013 Saltire Software. All rights reserved. Information in ...»

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3. Either click the cursor to place the reflection line on the screen, adjust the angle and click again, or select an existing line as the reflection line.

A copy of your selected geometry will appear on the other side of the reflection line.

–  –  –

Notice all points on the reflected geometry are written as "prime", i.e. A becomes A'. If you reflect the geometry again, A' becomes A".

Translation You can translate any subset of your diagram with a translation vector.

Here are the steps:

–  –  –

2. Click the Translation tool in the Construct toolbox, or select Translation from the Construct menu.

3. Click the cursor to draw the end point of your translation vector and move the cursor to establish the length and angle of the translation.

Click again to finish the vector.

The translated geometry appears. You can adjust the position of the translation by clicking and dragging the tip of the vector.

Notice all points on the translated geometry are written as "prime", i.e. A becomes A'. If you translate this geometry again, A' becomes A".

Rotation You can rotate any subset of your diagram about a point. Here are the

steps:

© 2013 Saltire Software 104 Geometry Expressions Manual

–  –  –

2. Click the Rotation tool in the Construct toolbox, or select Rotation from the Construct menu.

3. Click the screen to place your rotation point.

4. In the data entry box presented, enter the angle of rotation.

The rotation of the selected geometry appears.

Notice all points on the rotated geometry are written as "prime", i.e. A becomes A'. If you rotate this geometry again, A' becomes A".

Dilation You can dilate any subset of your diagram from a point. Here are the

steps:

–  –  –

2. Click the Dilation tool in the Construct toolbox, or select Dilation from the Construct menu.

3. Click the cursor on your dilation point.

4. In the data entry box presented, enter the dilation factor.

The dilated geometry appears.

Notice all points on the dilated geometry are written as "prime", i.e. A becomes A'. If you dilate this geometry again, A' becomes A".

–  –  –

Locus of Points / Envelope You can construct a locus of points or envelope from a selected point or line, by defining a range for some constraint in the drawing. Just follow

these easy steps:

1. Select the point on the drawing that will form the locus (point B in the example below); select a line, line segment, or vector to form an envelope.

–  –  –

3. You need a parameter to drive the motion to create the locus. Click the arrow key to the right of the Parametric Variable window to select from a list of all variables in the drawing. (If you entered the needed constraint in real terms, Cancel the trace and change the constraint to a variable by double-clicking it in the drawing window.)

4. Simply fill in the values for the appropriate variable and click the Ok button.

© 2013 Saltire Software 106 Geometry Expressions Manual Note: The locus only works if the figure's position is fixed i.e. a point in a triangle will not work as a locus unless the location of the other two points are fixed with coordinate constraints.

To adjust the range of the locus, double-click the locus to edit the dialog.

The following example shows an envelope of the line DE. We use the Point proportional along curve constraint and the parameter t to position the points D and E (D is (1-t) along line AC and E is t along line AB). In the Edit Locus dialog, we create the envelope from parameter t as it ranges from 0 to 1.

–  –  –

You can replay the creation of the envelope with the Animation tools.

Select variable t and click the Play button.

Trace You can Trace the movement of one or a group of drawing objects. You can create string art drawings and see how an envelope curve is formed.

Here are the steps:

1. Select the drawing objects to be traced.

2. Click the Trace tool and the Edit Trace dialog pops up.

3. You need a parameter to drive the motion of the trace. Click the arrow key to the right of the Parametric Variable window to select © 2013 Saltire Software 108 Geometry Expressions Manual from a list of all variables in the drawing. (If you entered the needed constraint in real terms, Cancel the trace and change your constraint to a variable by double-clicking it in the drawing window.)

4. Simply fill in the values for the appropriate variable and click the OK button.

To adjust the range or number of traces, double-click one of the traces to edit the dialog.

–  –  –

Area Under the Arc The Area Under Arc function is found only in the Construct menu at the

top of the main window. Here are the steps:

1. Select an arc drawn over a function.

2. Select Construct / Area Under Arc.

A filled area is created between the arc and the X axis. The necessary lines and points containing the area are automatically added.

Drag the handles of the original function and the area under the curve changes accordingly.

© 2013 Saltire Software 110 Geometry Expressions Manual This is a great way to illustrate the definite integral.

Note: this only works for arcs drawn on functions. For arcs drawn on conics, create the sides with the Draw / Line Segment, select all sides and use the Construct / Polygon tool. See also: instructions for creating curvilinear polygons.

Integral Calculus The Area Under Arc function is a great way to show problems with the definite integral.

Here are the steps for making the integral of the generic function f(x) over

the interval a to b:

1. Select the Draw / Function tool and enter f(x) in the Y= data entry box. Click OK.

2. Use the Draw / Arc tool to trace over some portion of the function.

3. Select the new arc - be careful not to select the whole function. Just the section of arc should be highlighted.

4. Choose Area Under Arc from the Construct menu. The area between the arc and the X axis will be filled.

5. For each endpoint of the arc, use the Constrain / Point proportional along curve tool to define the limits of the region, e.g.

–  –  –

Annotations Applying Annotations Annotations allow you to add constraint information to your drawing which isn't needed for it's construction. These might be constraints that cause the geometry to be over constrained, but you might want to give the viewer some additional information. This feature can be very helpful for making up texts or worksheets.

Annotate tools are similar to Text in that they have no influence on the geometry engine, even though they are placed exactly like the Constrain tools.

Since annotations may look identical to constraints, use the icon from icon

–  –  –

© 2013 Saltire Software 112 Geometry Expressions Manual Annotations. The icon is a toggle; to turn off the marks, click it again.

The Symbols and Annotation Symbols toolboxes are both available to the Annotate tools.

Here are the Annotate tools and the object(s) to preselect:

–  –  –

Distance / Length Annotation Length annotations may be applied to any line, segment, polygon side or vector. Distance annotations are available between two points, or the perpendicular distance between a point and a line, segment, polygon side,

or vector. Use these steps:

1. Select the line segment, or point and line, or pair of points as described above. When you make your selection, the drawing objects will be highlighted and the Distance / Length icon will light up.

2. Click the icon

3. Enter the distance information. There are no format restrictions. You can use Symbols and Annotation Symbols in the entry. Press enter when you're done.

You can click the annotation and drag it to adjust its placement in the drawing.

To annotate a circle's radius use these steps:

1. Select the circle. The Annotate / Radius icon will light up and the circle will be highlighted.

2. Click the icon

3. Enter the radius information. There are no format restrictions. You can use Symbols and Annotation Symbols in the entry. Press enter © 2013 Saltire Software 114 Geometry Expressions Manual

–  –  –

You can click the annotation and drag it to adjust its placement in the drawing.

Perpendicular Annotation The perpendicular annotation inserts the perpendicular mark between any two of these line types: line, segment, polygon side or vector. Use these

steps:

1. Select two of any line, segment, vector, or polygon side. The Annotate / Perpendicular icon will light up when both lines are highlighted.

2. Click the icon.

Note: Placing this annotation between two lines does not change the relative position of the lines or prevent the lines from changing their relative position as it does with the Constrain tool of the same name.

This annotation is very useful if you are displaying a 3-d object. In this example we imported the Triangular Prism from the Solids folder in the Figure Gallery.

–  –  –

Which Side to Annotate?

Sometimes when identifying angles, the annotation falls on the wrong one.

In the example below, we wanted BDC, not BDA. Just click the cursor over the annotation arrow and drag it to the other side, then release the mouse button - done!

© 2013 Saltire Software 116 Geometry Expressions Manual Direction Annotation

To annotate a line's direction use these steps:

1. Select a line, segment, vector, or polygon side. The Annotate / Direction icon will light up and the line will be highlighted.

2. Click the icon

3. Enter the direction information. There are no format restrictions.

You can use Symbols and Annotation Symbols in the entry. Press enter when you're done.

You can click the annotation and drag it to adjust its placement in the drawing.

Slope Annotation

To annotate a line's slope use these steps:

1. Select a line, segment, vector, or polygon side. The Annotate /

–  –  –

Slope icon will light up and the line will be highlighted.

2. Click the icon

3. Enter the slope information. There are no format restrictions. You can use Symbols and Annotation Symbols in the entry. Press enter when you're done.

You can click the annotation and drag it to adjust its placement in the drawing.

Coordinate Annotation

To annotate an point's coordinates use these steps:

1. Select point. The Annotate / Coordinate icon will light up and the point will be highlighted.

2. Click the icon

3. Enter the coordinate information. There are no format restrictions.

You can use Symbols and Annotation Symbols in the entry. Press enter when you're done.

You can click the annotation and drag it to adjust its placement in the drawing.

Coefficients Annotation

To annotate a vector's coefficients use these steps:

© 2013 Saltire Software 118 Geometry Expressions Manual

1. Select vector. The Annotate / Coefficients icon will light up and the vector will be highlighted.

2. Click the icon

3. Enter the vector's coefficients. There are no format restrictions. You can use Symbols and Annotation Symbols in the entry. Press enter when you're done.

You can click the annotation and drag it to adjust its placement in the drawing.

Congruent Annotation Place a congruent mark on any of the linear drawing elements: lines,

segments, vectors, or polygon sides. Use these steps:

1. Select any line, segment, vector, or polygon side. The Annotate / Congruent icon will light up when both lines are highlighted.

2. Click the icon.

–  –  –

Note: Placing this annotation on lines does not change the relative lengths of the lines or keep them the same length as it does with the Constrain tool of the same name.

Congruent Angle Annotation Place a congruent mark between pairs of linear drawing elements: lines,

segments, vectors, or polygon sides. Use these steps:

1. Select two lines, segments, vectors, or polygon sides. The Annotate / Congruent Angle icon will light up when both lines are highlighted.

2. Click the icon.

© 2013 Saltire Software 120 Geometry Expressions Manual

You can change the arc count with these steps:

1. Select the annotation.

2. Right click to invoke the selection Context menu.

3. Select Tic/Arc Count from the menu and click the desired number.

–  –  –

Parallel Annotation Place a parallel mark on any of the linear drawing elements: lines,

segments, vectors, or polygon sides. Use these steps:

–  –  –

Note: Placing this annotation on lines does not change the relative position of the lines or prevent the lines from changing their relative position as it does with the Constrain tool of the same name.

Expression Annotation The Annotate / Expression command is useful for placing a line of mathematics anywhere in your drawing.

© 2013 Saltire Software 122 Geometry Expressions Manual

1. Click the icon.

2. Move the cursor to the position where you want to place the expression and click.

3. Enter the expression. There are no format restrictions. You can use Symbols and Annotation Symbols in the entry. Press enter when you're done.

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