You will use this worksheet to perform four basic constructions: [br] 1) Copy a segment[br] 2) Bisect a segment[br] 3) Copy an angle[br] 4) Bisect an angle

In the window below, use the tools to copy the given segment. Use the Distance tool to show that the segments are the same length.[br][br]T[size=85][size=100]o construct a copy of line segment AB, complete these steps:[br][b]Step 1: Create a new point[/b][list][*]Select the [b]Point tool[/b].[/*][*]Click anywhere on the workspace to create a new point.[/*][*]This will be the starting point of your copied segment.[br][/*][/list][b]Step 2: Set the compass to the original segment[/b][list][*]Select the [b]Compass tool[/b].[/*][*]Click on the two endpoints of the original segment.[/*][*]This sets the compass width equal to the length of the original segment.[br][/*][/list][b]Step 3: Place the circle at the new point[/b][list][*]Move your cursor to the new point you just created and click.[br][/*][*]A circle will appear, centered at that point, with radius equal to the original segment.[br][/*][/list][b]Step 4: Mark the intersection point[/b][list][*]Select the [b]Point tool[/b].[/*][*]Click anywhere on the circle to create a new point on its circumference.[br][/*][/list][b]Step 5: Draw the copied segment[/b][list][*]Select the [b]Segment tool[/b].[/*][*]Click on the new starting point (from Step 1) and the new point on the circle (from Step 4).[/*][*]A new line segment will be constructed.[br][/*][/list][b]Step 6: Measure both segments[/b][list][*]Select the [b]Distance or Length tool[/b].[/*][*]Click on the original segment.[/*][*]Then click on the new segment you just created.[/*][*]The lengths of both segments will appear.[/*][/list][/size][/size]

In the window below, use the tools to construct a perpendicular bisector. Use the distance tool to show that the two pieces of the segment are the same length.[br][br]Segment Bisector StepsAny segment, line or plane that intersects a segment at its midpoint is called a segment bisector.[br][br][b]Step 1: Create a helper point[/b][list][*]Select the [b]Point tool[/b].[/*][*]Place a point anywhere between the two endpoints of the segment (not in the exact middle — closer to one endpoint).[/*][*]This helper point is only used to set the compass radius.[br][/*][/list][b]Step 2: Set the compass using the farther endpoint[/b][list][*]Select the [b]Compass tool[/b].[/*][*]Click on the [b]endpoint that is farther from the helper point[/b], then click on the [b]helper point[/b].[/*][*]This sets the compass radius to more than half the length of the segment.[br][/*][/list][b]Step 3: Drop a circle at the first endpoint[/b][list][*]Move your cursor back to the [b]first endpoint of the segment[/b] and click to drop a circle.[br][/*][/list][b]Step 4: Repeat Step 2 and place a circle at the second endpoint[/b][list][*]Again, with the [b]Compass tool[/b], click on the [b]endpoint farther from the helper point[/b], then click on the [b]helper point[/b] (this resets the same radius).[/*][*]Now move your cursor to the [b]second endpoint of the segment[/b] and click to drop another circle.[/*][*]You should now see two overlapping circles above and below the segment.[br][/*][/list][b]Step 5: Mark the circle intersections[/b][list][*]Select the [b]Point tool[/b].[/*][*]Click the two points where the circles intersect (one above and one below the segment).[br][/*][/list][b]Step 6: Connect the intersections[/b][list][*]Select the [b]Line tool[/b] (or Segment tool).[br][/*][*]Draw a line through the two intersection points.[/*][*]This line crosses the original segment at its midpoint.[br][/*][/list][b]Step 7: Mark the midpoint[/b][list][*]Select the [b]Point tool[/b].[/*][*]Click on the intersection where the new line crosses the original segment.[/*][*]This point is the [b]midpoint[/b] of the segment.[br][/*][/list][b]Step 8: Check your construction[/b][list][*]Select the [b]Distance or Length tool[/b].[/*][*]Measure from the first endpoint to the midpoint.[/*][*]Then measure from the midpoint to the second endpoint.[/*][*]The two distances should be equal.[/*][/list]

In the window below, use the tools to construct an angle congruent to the one given. Use the angle measurement tool to show that the angles have the same measure.[br][br][b]Step 1: Draw the new ray[/b][br][list][*]Select the [b]Ray tool[/b].[/*][*]Click anywhere on the workspace (this point will eventually be the vertex of your copied angle).[/*][*]Click a second point to extend the ray.[/*][*]Now you have a starting ray for your new angle.[br][/*][/list][b]Step 2: Create an arc across the original angle[/b][list][*]Select the [b]Compass tool[/b].[/*][*]Click on the [b]vertex of the original angle[/b], then click on a point on one of its rays (this sets the compass width).[/*][*]Move your cursor back to the [b]vertex of the original angle[/b] and click to drop a circle.[/*][*]This circle crosses [b]both rays of the original angle[/b].[/*][*]Mark both intersection points where the circle meets the rays.[br][/*][/list][b]Step 3: Copy that arc onto the new ray[/b][list][*]With the Compass tool still active, click the [b]vertex of the original angle[/b] and then one of the arc’s intersection points (from Step 2).[/*][*]This saves the arc size.[/*][*]Now move your cursor to the [b]endpoint of the new ray[/b] you created in Step 1 and click to drop a circle.[/*][*]This circle crosses the new ray.[/*][*]Mark the intersection point on the ray.[br][/*][/list][b]Step 4: Transfer the distance between the two original intersections[/b][list][*]Select the [b]Compass tool[/b] again.[/*][*]Click on the [b]two intersection points[/b] from Step 2 (the ones on the original angle’s rays).[/*][*]This sets the compass width equal to the “gap” between the rays.[/*][*]Move your cursor to the [b]intersection point on the new ray[/b] from Step 3 and click to drop a circle.[/*][*]This new circle crosses the circle you dropped in Step 3.[/*][*]Mark their intersection point.[br][/*][/list][b]Step 5: Draw the second ray of the new angle[/b][list][*]Select the [b]Ray tool[/b].[/*][*]Click the [b]endpoint of the new ray[/b] (from Step 1).[/*][*]Then click the [b]intersection point[/b] you marked in Step 4.[/*][*]This creates the second ray of your copied angle.[br][/*][/list][b]Step 6: Check with the Angle tool[/b][list][*]Select the [b]Angle tool[/b].[/*][*]Click the three points that make up the original angle (ray point – vertex – ray point). GeoGebra will display its measure.[/*][*]Then click the three points of the new angle (ray point – new vertex – ray point).[/*][*]Compare the two measures.[/*][*]If they are the same, the new angle is [b]congruent[/b] to the original angle.[/*][/list]

In the window below, use the tools to bisect the given angle. Use the angle measuring tool to show that the two angles are congruent.[br][br][b]Step 1: Create an arc across the angle[/b][list][*]Select the [b]Compass tool[/b].[/*][*]Click on the vertex of the angle, then on a point on one of its rays (this sets the compass width).[/*][*]Move your cursor back to the [b]vertex of the angle[/b] and click to drop a circle.[/*][*]The circle should intersect [b]both rays[/b] of the angle.[/*][*]Mark the two intersection points.[br][/*][/list][b]Step 2: Set the compass to the distance between intersections[/b][list][*]With the [b]Compass tool[/b], click on one of the intersection points and then the other.[/*][*]This sets the compass width equal to the distance between them.[br][/*][/list][b]Step 3: Place a circle at the first intersection point[/b][list][*]Move your cursor back to the [b]first intersection point[/b] and click to drop a circle.[br][/*][/list][b]Step 4: Place a circle at the second intersection point[/b][list][*]Repeat Step 2 (click the same two intersection points again to keep the radius).[/*][*]Move your cursor back to the [b]second intersection point[/b] and click to drop another circle.[/*][*]The two circles should overlap inside the angle.[br][/*][/list][b]Step 5: Mark the circle intersection inside the angle[/b][list][*]Select the [b]Point tool[/b].[/*][*]Click the point where the two circles intersect inside the angle.[br][/*][/list][b]Step 6: Draw the angle bisector[/b][list][*]Select the [b]Ray tool[/b].[/*][*]Click the [b]vertex of the angle[/b], then click the intersection point from Step 5.[/*][*]A ray will appear that divides the angle into two equal parts.[br][/*][/list][b]Step 7: Check your construction[/b][list][*]Select the [b]Angle tool[/b].[/*][*]Measure one side of the bisected angle (ray–vertex–bisector).[/*][*]Then measure the other side (bisector–vertex–ray).[/*][*]If both measures are equal, your construction is correct.[/*][/list]

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