Ruler and Compass Construction Appendix: Tangents to Circles.



Section 0: The Basics of SketchPad

 The basic geometric objects that SketchPad knows how to draw are points, lines, and circles. You can access these objects using the tool bar on the left side of the screen. The functions of these buttons are as follows.


Section 1: Tangency and Perpendicularity

 A tangent line to a circle is a line which touches the circle in only one place. This section is devoted to exploring the relationship between tangency of a line to a circle and perpendicularity with the radius of the circle.

Use the Circle tool to draw a circle on the screen. Label the center as A. Draw a line on the screen which intersects the circle in two places. Click on the Text tool. Click on the line. This ought to produce the label, j.

Note: To change the label, use the Text tool and double click on the label that you want to change. A box will appear where you can change the label.

Using the Selection tool, click anywhere on the screen where you have not drawn anything. Click on j, hold down the shift key, and click on the circle. Go to the Construct menu and pull down to Point At Intersection. This will mark the points where the circle and line intersect. Label them as C and D. 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 C. Go to the Construct menu and pull down to Segment. This will draw a line segment between A and C.

Note: If you need to erase something that you have drawn, click on the object with the Selection tool and either hit the backspace key or go to the Edit menu and pull down to Clear. If you want to erase a group of objects, use the Selection tool to drag out a box containing all the objects that you want to erase and then hit Backspace or use the Clear function.

Using the Selection tool, click anywhere on the screen where you have not drawn anything. Click on A, hold down the shift key, click on C, hold down the shift key, and click on D. Go to the Measure menu and pull down to Angle. This will measure the angle, ACD. Note that this angle is not a right angle. 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 Perpendicular Line. This will draw a line perpendicular to j passing through A.

Question: How can you use this line and the first point of intersection to find the second point of intersection? If you reflect the drawing about this line, where does the point of intersection have to go to?. Once you have this relationship, prove the following: Let j be a line intersecting a circle with center A at a point C. If j is not perpendicular to the line from A to C, then there is a second point of intersection (j is not tangent to the circle). Put the above statement into the positive form (the contrapositive). That is: If j is tangent to the circle at C, then ...























Create a new sketch. Draw a circle on the screen and label the center as A. Use the Point tool to draw a point outside the circle. Construct a line segment from the center to this point using the Segment function (see above for how to do this). Label this line as j. Mark the point of intersection of j (see above for how to do this) and the circle and label it as D. Using the Selection tool, click anywhere on the screen where you have not drawn anything. Click on D, hold down the shift key, and click on j. Go to the Construct menu and pull down to Perpendicular Line. Label this line as k. Use the Point tool to draw a point on this line. Label it as E. Using the Selection tool, click anywhere on the screen. Click on A, hold down the shift key, and click on E. Go to the Construct menu and pull down to Segment. This will drawn a line from A to E.

Question: Using the Selection tool, click anywhere on the screen where you have not drawn anything. Click and hold down on E. Move the mouse around. What is the relationship between the lengths AD, AE, and DE? Can you express one of these lengths in terms of the other two? If k isn't tangent to the circle, and E is the other point of intersection, what does your relationship tell you about the distance DE? Is it possible for k to interesect the circle in two distinct points?























You have now proved the following:

Theorem: A line is tangent to a circle if and only if it is perpendicular to the radius at the point of contact.

Section 2: Constructing Tangent Lines to Circles with a Ruler and Compass

 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 point, draw a straight line 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. Using these functions, one can construct perpendicular bisectors of segments and bisectors of angles. So, we will also allow the Perpendicular Line function. It is impossible to trisect an arbitrary angle using only a Ruler and Compass.

Create a new sketch. Draw a circle on the screen. Label the center as A. Draw a point outside the circle and label it as C. We want to construct, using a Ruler and Compass, a line which is tangent to the circle and passes through C.

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 C. Go to the Construct menu and pull down Circle by Center and Point. This will draw a circle whose center is A and with radius the distance from A to C. Construct a line segment from A to C using the Segment function (see section 1 if you need a reminder). Mark the point of intersection between this line and the inner circle using the Point at Intersection function. Label it as D. Construct a line perpendicular to your line which passes through D using the Perpendicular Line function. Mark the points of intersection of this line with the outer circle using the Point At Intersection function. Label one of the points as E. Construct a line segment from A to E using the Segment function. Mark the point of intersection of this line with the inner circle using the Point at Intersection function. Label it as F. Construct a line segment from C to F using the Segment function.

Using the Selection tool, click anywhere on the screen where you have not drawn anything. Click on C, hold down the shift key, click on F, hold down the shift key, and click on A. Go to the Measure menu and pull down to Angle. This will measure the angle CFA.

Question: Using the Selection tool, click and hold down on C. Move C around and see how the angle changes (if at all). What does your result from section 1 tell you about the tangency of the line from C to F with the inner circle? Find a triangle which is similar to the triangle CFA. Use this and your result from section 1 to show that your construction always works. Is there a second line which is tangent to the circle and passes through C? How can you find it?