## Tangent Lines to Circles

Today's lesson we are going to find the tangent line from a cicle to abitrary point outside of the circle. This construction will be done using only a compass and a straight edge.

You can follow the construction with the sketches below. The markings in red are the construction each step.

# Step 1

Start by drawing a cirlce with your compass. Label the center A and the circle c1. Next, select an arbitrary point outside of the circle, label this P.

# Step 2

Using your straight edge, connect the center A with the point outside the circle, P. This gives us the line segment AP. Using your compass, conctruct a perperndicular bisector of AP (This is shown below by the dashed line). The intersection of the two lines is the midpoint of AP. Label this point M.

# Step 3

Measure the distance from M to P with your compass. Use this distance as the radius,with your compass, construct a circle with M as the center. Your cicle should go through points A and P as shouwn below. Label this circle c2. Label the intersection points of c1 and c2 point H and point G.

# Step 4

Finally, using your straight edge, draw a line through points P and H. Do the same with points P and G (extend the lines through the points). These lines of intersection are the tangents to our original circle C1.

You now have a circle and a line tangent to the cicle and a point outside of the cirlce.