Step-by-step Basic Constructions Practice

Author:ANASEINI BARTHELEMY, Sandra Miller

Purpose

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

1) Copy a segment

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. To construct a copy of line segment AB, complete these steps: Step 1: Create a new point

Select the Point tool.
Click anywhere on the workspace to create a new point.
This will be the starting point of your copied segment.

Step 2: Set the compass to the original segment

Select the Compass tool.
Click on the two endpoints of the original segment.
This sets the compass width equal to the length of the original segment.

Step 3: Place the circle at the new point

Move your cursor to the new point you just created and click.
A circle will appear, centered at that point, with radius equal to the original segment.

Step 4: Mark the intersection point

Select the Point tool.
Click anywhere on the circle to create a new point on its circumference.

Step 5: Draw the copied segment

Select the Segment tool.
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.

Step 6: Measure both segments

Select the Distance or Length tool.
Click on the original segment.
Then click on the new segment you just created.
The lengths of both segments will appear.

2) Bisect a segment

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. Segment Bisector StepsAny segment, line or plane that intersects a segment at its midpoint is called a segment bisector. Step 1: Create a helper point

Select the Point tool.
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.

Step 2: Set the compass using the farther endpoint

Select the Compass tool.
Click on the endpoint that is farther from the helper point, then click on the helper point.
This sets the compass radius to more than half the length of the segment.

Step 3: Drop a circle at the first endpoint

Move your cursor back to the first endpoint of the segment and click to drop a circle.

Step 4: Repeat Step 2 and place a circle at the second endpoint

Again, with the Compass tool, click on the endpoint farther from the helper point, then click on the helper point (this resets the same radius).
Now move your cursor to the second endpoint of the segment and click to drop another circle.
You should now see two overlapping circles above and below the segment.

Step 5: Mark the circle intersections

Select the Point tool.
Click the two points where the circles intersect (one above and one below the segment).

Step 6: Connect the intersections

Select the Line tool (or Segment tool).
Draw a line through the two intersection points.
This line crosses the original segment at its midpoint.

Step 7: Mark the midpoint

Select the Point tool.
Click on the intersection where the new line crosses the original segment.
This point is the midpoint of the segment.

Step 8: Check your construction

Select the Distance or Length tool.
Measure from the first endpoint to the midpoint.
Then measure from the midpoint to the second endpoint.
The two distances should be equal.

3) Copy an angle

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. Step 1: Draw the new ray

Select the Ray tool.
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.

Step 2: Create an arc across the original angle

Select the Compass tool.
Click on the vertex of the original angle, then click on a point on one of its rays (this sets the compass width).
Move your cursor back to the vertex of the original angle and click to drop a circle.
This circle crosses both rays of the original angle.
Mark both intersection points where the circle meets the rays.

Step 3: Copy that arc onto the new ray

With the Compass tool still active, click the vertex of the original angle and then one of the arc’s intersection points (from Step 2).
This saves the arc size.
Now move your cursor to the endpoint of the new ray you created in Step 1 and click to drop a circle.
This circle crosses the new ray.
Mark the intersection point on the ray.

Step 4: Transfer the distance between the two original intersections

Select the Compass tool again.
Click on the two intersection points 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 intersection point on the new ray from Step 3 and click to drop a circle.
This new circle crosses the circle you dropped in Step 3.
Mark their intersection point.

Step 5: Draw the second ray of the new angle

Select the Ray tool.
Click the endpoint of the new ray (from Step 1).
Then click the intersection point you marked in Step 4.
This creates the second ray of your copied angle.

Step 6: Check with the Angle tool

Select the Angle tool.
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 congruent to the original angle.

4) Bisect an angle

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. Step 1: Create an arc across the angle

Select the Compass tool.
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 vertex of the angle and click to drop a circle.
The circle should intersect both rays of the angle.
Mark the two intersection points.

Step 2: Set the compass to the distance between intersections

With the Compass tool, click on one of the intersection points and then the other.
This sets the compass width equal to the distance between them.

Step 3: Place a circle at the first intersection point

Move your cursor back to the first intersection point and click to drop a circle.

Step 4: Place a circle at the second intersection point

Repeat Step 2 (click the same two intersection points again to keep the radius).
Move your cursor back to the second intersection point and click to drop another circle.
The two circles should overlap inside the angle.

Step 5: Mark the circle intersection inside the angle

Select the Point tool.
Click the point where the two circles intersect inside the angle.

Step 6: Draw the angle bisector

Select the Ray tool.
Click the vertex of the angle, then click the intersection point from Step 5.
A ray will appear that divides the angle into two equal parts.

Step 7: Check your construction

Select the Angle tool.
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.

You're done!

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