 Construct: Tangents to Circles MathBitsNotebook.com Terms of Use   Contact Person: Donna Roberts  Use only your compass and straight edge when drawing a construction. No free-hand drawing!

 Tangent to Circle at Point ON Circle
 This construction is an easy one if you remember that in a circle, a radius drawn to the point of tangency is perpendicular to the tangent. Use the construction: construct a perpendicular to a line from a point on the line. This construction is simply a variation of a construction you already know how to draw. Given: Circle O Construct: a tangent to circle at a point on the circle STEPS: 1. If a point on the circle is not given, draw any radius and label P. If a point is already given on the circle, connect the point to the center of the circle to form a radius. 2. Extend the radius past the circle. 3. Construct a perpendicular to the radius line at point P.  Tangent to Circle from Point OUTSIDE Circle (G.C.4)
 This construction requires a bit more work. Picture in your mind what these tangents to a circle will look like. The diagram at the right shows two tangents from point P. You need not draw the two tangents, but the two of them may remind you of how this construction will look, as the construction creates two possible tangents. Given: Circle O Construct: a tangent to circle O from P STEPS: 1. Connect O to P. 2. Construct bisector of . 3. Place compass point at midpoint of and stretch span to O or P. 4. Draw circle. 5. Connect P to where the two circles intersect to create tangents. Why does this work? When the construction is finished, connect O to A to form a radius of circle O. This radius also forms ΔOAP in circle M. Since is the diameter of circle M, ∠OAP is an angle inscribed in a semicircle, making it a right angle. Since ∠OAP is a right angle, . In circle O, we now have the radius perpendicular to a line passing through a point on the circle (A), making a tangent to circle O.  Finding the Center of a Circle
 What do you do when a construction problem involving circles, gives you a starting circle such as that shown at the right? There is NO CENTER indicated on the circle. Unfortunately, you can NOT plot your best guess of where you think the center may be located. If you encounter this situation, you will have to CONSTRUCT the location of the center. Given: Circle with no center indicated Construct: locate the center of the circle STEPS: 1. Draw an inscribed angle (an angle with its vertex on the circle and sides terminating on the circle). (This construction also works if you draw two chords instead of the inscribed angle. Drawing the ∠ keeps the chords positioned to more clearly find the center.) 2. Bisect each side of the angle (or chord). 3. The point where the bisectors intersect is the center of the circle.
 Why this works: There is a theorem that states, "in a circle, the perpendicular bisector of a chord passes through the center of the circle". The diagram at the right shows how to support this theorem with congruent triangles.  