Lab 2: Ruler and Compass Constructions



Section 1: Measuring in SketchPad

 SketchPad has several built in measurement features. The ones we are interested in are: Distance, Length, and Angle.

Task: Draw two points on the screen. Click on the Text tool (the little hand) and then click on the first point. This will produce some label for the point. If it isn't A, double click on the label that was produced. A box will appear where you can change the label to A. Label the other point as B. Using the Selection tool (the arrow), click anywhere on the screen where you have not drawn anything. Click on A, hold down the shift key, and click on B. Go to the Measure menu and pull down to Distance. This will display the distance between the two points.

Task: Using the Selection tool, click anywhere on the screen where you have not drawn anything. Select both points. Go to the Construct menu and pull down to Segment. This will draw a line segment between A and B. Use the Selection tool to select this new line. Go to the Measure menu and select Length. This will produce the length of the line segment.

Task: Draw another point on the screen which is not on the old line segment. Label it as C. Construct a line segment between A and C. Construct a line segment between B and C. You should now have a triangle. Using the Selection tool, click anywhere on the screen which doesn't have something drawn on it. Now, click on point A, hold down the shift key, click on point B, hold down the shift key, and click on point C. Go to the Measure menu and pull down the Angle. This will measure the angle between the lines AB and BC.

Section 2: Ruler and Compass Constructions in SketchPad

 A Ruler and Compass construction is a drawing (construction) that you create using only a straight edge and a compass. This means you can draw a straightline between two points, draw a circle given a center and a radius, and find points of intersection. The word "Ruler" is more traditional than anything else. In fact, you are not allowed to use any measuring tools in a Ruler and Compass construction. In SketchPad, a Ruler and Compass construction is one that you perform using only the Point tool, the Segment function, the Circle by Center and Radius function, the Circle by Center and Point function, and the Point at Intersection function.

Create a new sketch. We have already met the Segment function. We'll now see how to use the other functions. Draw a point on the screen and label it A. Draw another point (not too far away from A) on the screen and label it B. Construct a line segment (using the Segment function) between A and B and label it j. Using the Selection tool, click anywhere on the screen where you have not drawn anything. Click on A, hold down the shift key, and click on j. Go to the Construct menu and pull down to Circle by Center and Radius. This will draw a circle whose center is A and whose radius is j. Using the Selection tool, click anywhere on the screen where you have not drawn anything. Click on B, hold down the shift key, and click on A. Go to the Construct menu and pull down to Circle by Center and Point. This will draw a circle whose center is B and whose radius is the distance from B to A. Using the Selection Tool, click anywhere on the screen where you haven't drawn anything. Click on one of the circles, hold down the shift key, and click on the other circle. Go to the Construct menu and pull down to Point at Intersection. This will mark the points at which the two circles intersect. We will call this set up the Basic Construction.

Question: Why did you get two points of intersection? Does this work on a sphere?







Section 3: Constructing Equilateral Triangles with a Ruler and Compass

 An equilateral triangle is a triangle all of whose sides have the same length.

Task: Starting with the Basic Construction, construct an equilateral triangle using a Ruler and Compass (see the start of section 2 is you have forgotten what this means). Measure the lengths of the sides and angles of your triangle to verify that your construction works. Make sure to move the triangle around somewhat to see what happens to the lengths of the sides and the angles.







Question: Why does your construction always work in the plane? Why does it give you an equilateral triangle?







Section 4: Constructing the Perpendicular Bisector of a Segment With a Ruler and Compass

 Create a new sketch. Draw the Basic Construction from section 2.

Task: Try to construct the perpendicular bisector of the line segment, j (the line between the two points in the construction from section 2), with a ruler and compass. Think about the construction of equilateral triangles from section 3. Once you have a guess, verify it by measuring the relevant angles and lengths. Move the drawing around a bit. Make sure that your construction still works.

Note: You will have to construct the point of intersection between j and the perpendicular bisector that you have constructed in order to measure the appropriate distances and angles.

In order to answer the question below, you will need the following proposition, which we proved in class when we investigated the Isoceles Triangle Theorem

Proposition: For an isoceles triangle, the bisector of the angle formed by the two edges of equal length is the perpendicular bisector of the remaining edge.

Task: In the space below, write a proof that the line you constructed above is, in fact, the perpendicular bisector. When you have finished your proof, find another group that has completed this section and exchange proofs. Read over the other group's proof and check it for correctness. When both groups have finished checking the other's proofs, discuss any errors or changes that should be made.















































Section 5: Constructing the Bisector of an Angle With a Ruler and Compass

 Create a new sketch. In section 4 we constructed the perpendicular bisector of a line segment using only a ruler and compass. In this section, we'll construct the bisector of an angle using a ruler and compass (same rules apply as in section 4).

Begin by drawing a point on the screen. Label it as A. Click, and hold down, on the Line tool. A box will slide to the right with other tools that you can choose. Move the mouse to the one which looks like a ray (has an arrowhead on it). Click on A (and hold down) and move the mouse a bit. A ray eminating from A will appear on the screen. Position it as you wish and let go of the mouse button. Draw another ray coming from A (you ought to have something that looks like a hinge). This will form the angle that you are to bisect. When you are done drawing the rays, click and hold down on the Line tool. The box will slide to the left again. Select the left most box (the segment icon). Draw a point on one of the rays. Use the Circle by Center and Point function to construct a circle with center A and radius the distance between A and the point you just drew. Find the point of intersection between the other ray and the circle.

Task: Starting with this set up, use only ruler and compass constructions to construct the angle bisector of the hinge. You'll use methods similar to those in section 4. Once you have what you think is the bisector, measure the appropriate angles to verify that your construction works. Move the drawing around and see if it still works.

Question: Write down a proof that your construction does in fact give the angle bisector. You may want to use your result from section 4. When you have finished your proof, find another group that has completed this section and exchange proofs. Read over the other group's proof and check it for correctness. When both groups have finished checking the other's proofs, discuss any errors or changes that should be made.















































Section 6: Trisecting an Angle

 Start on this section only if you have fully completed sections 4 and 5!. In section 5 we constructed the bisector of an angle using only a Ruler and Compass. In this section, we will try to trisect an angle using only a Ruler and Compass. See above if you have forgotten what this means and what functions you can use in SketchPad.

Begin by drawing a point on the screen. Label it as A. Click, and hold down, on the Line tool. A box will slide to the right with other tools that you can choose. Move the mouse to the one which looks like a ray (has an arrowhead on it). Click on A (and hold down) and move the mouse a bit. A ray eminating from A will appear on the screen. Position it as you wish and let go of the mouse button. Label this ray as j. Draw another ray coming from A (you ought to have something that looks like a hinge) and label it as k. This will form the angle that you are to trisect. When you are done drawing the rays, click and hold down on the Line tool. The box will slide to the left again. Select the left most box (the segment icon).

Task: Using only a Ruler and Compass, trisect this angle. Once you have a guess as to how to do this, measure the appropriate angles to verify that you have trisected the angle. If you are unable to trisect the angle, try to find an angle that you can trisect.