Prerequisite Skills This lesson requires the use of the following skills: understanding the relationship between perpendicular lines using a compass and a straightedge constructing a perpendicular bisector of a line segment Introduction Tangent lines are useful in calculating distances as well as diagramming in the professions of construction, architecture, and landscaping. Geometry construction tools can be used to create lines tangent to a circle. s with other constructions, the only tools you are allowed to use are a compass and a straightedge, a reflective device and a straightedge, or patty paper and a straightedge. You may be tempted to measure angles or lengths, but remember, this is not allowed with constructions. Key oncepts If a line is tangent to a circle, it is perpendicular to the radius drawn to the point of tangency, the only point at which a line and a circle intersect. xactly one tangent line can be constructed by using construction tools to create a line perpendicular to the radius at a point on the circle. U3-139
onstructing a Tangent at a Point on a ircle Using a ompass 1. Use a straightedge to draw a ray from center O through the given point P. e sure the ray extends past point P. 2. onstruct the line perpendicular to OP at point P. This is the same procedure as constructing a perpendicular line to a point on a line. a. Put the sharp point of the compass on P and open the compass less wide than the distance of OP. b. raw an arc on both sides of P on OP. Label the points of intersection and. c. Set the sharp point of the compass on. Open the compass wider than the distance of and make a large arc. d. Without changing your compass setting, put the sharp point of the compass on. Make a second large arc. It is important that the arcs intersect each other. 3. Use your straightedge to connect the points of intersection of the arcs. 4. Label the new line m. o not erase any of your markings. Line m is tangent to circle O at point P. It is also possible to construct a tangent line from an exterior point not on a circle. xactly two lines can be constructed that are tangent to the circle through an exterior point not on the circle. U3-140
If two segments are tangent to the same circle, and originate from the same exterior point, then the segments are congruent. onstructing a Tangent from an xterior Point Not on a ircle Using a ompass 1. To construct a line tangent to circle O from an exterior point not on the circle, first use a straightedge to draw a ray connecting center O and the given point R. 2. Find the midpoint of OR by constructing the perpendicular bisector. a. Put the sharp point of your compass on point O. Open the compass wider than half the distance of OR. b. Make a large arc intersecting OR. c. Without changing your compass setting, put the sharp point of the compass on point R. Make a second large arc. It is important that the arcs intersect each other. Label the points of intersection of the arcs as and. d. Use your straightedge to connect points and. e. The point where intersects OR is the midpoint of OR. Label this point F. 3. Put the sharp point of the compass on midpoint F and open the compass to point O. (continued) U3-141
4. Without changing the compass setting, draw an arc across the circle so it intersects the circle in two places. Label the points of intersection as G and H. 5. Use a straightedge to draw a line from point R to point G and a second line from point R to point H. o not erase any of your markings. RG and RH are tangent to circle O. If two circles do not intersect, they can share a tangent line, called a common tangent. Two circles that do not intersect have four common tangents. ommon tangents can be either internal or external. common internal tangent is a tangent that is common to two circles and intersects the segment joining the radii of the circles. common external tangent is a tangent that is common to two circles and does not intersect the segment joining the radii of the circles. U3-142
ommon rrors/misconceptions assuming that a radius and a line are perpendicular at the possible point of intersection simply by observation assuming two tangent lines are congruent by observation incorrectly changing the compass settings not making large enough arcs to find the points of intersection U3-143
Guided Practice 3.3.1 xample 1 Use a compass and a straightedge to construct tangent to circle at point. 1. raw a ray from center through point and extending beyond point. U3-144
2. Put the sharp point of the compass on point. Set it to any setting less than the length of, and then draw an arc on either side of, creating points and. 3. Put the sharp point of the compass on point and set it to a width greater than the distance of. Make a large arc intersecting. U3-145
4. Without changing the compass setting, put the sharp point of the compass on point and draw a second arc that intersects the first. Label the point of intersection with the arc drawn in step 3 as point. 5. raw a line connecting points and, creating tangent. o not erase any of your markings. is tangent to circle at point. U3-146
xample 2 Using the circle and tangent line from xample 1, construct two additional tangent lines, so that circle below will be inscribed in a triangle. 1. hoose a point, G, on circle. (Note: To highlight the essential ideas of this example, some features of the above diagram have been removed.) G U3-147
2. raw a ray from center to point G. G 3. Follow the process explained in xample 1 for constructing a tangent line through point G. G U3-148
4. hoose another point, H, on circle. raw a ray from center to point H, and follow the process explained in xample 1 to construct the third tangent line. e sure to draw the tangent lines long enough to intersect one another. H G o not erase any of your markings. ircle is inscribed in a triangle. U3-149
xample 3 Use a compass and a straightedge to construct the lines tangent to circle at point. 1. raw a ray connecting center and the given point. U3-150
2. Find the midpoint of by constructing the perpendicular bisector. Put the sharp point of your compass on point. Open the compass wider than half the distance of. Make a large arc intersecting. Without changing your compass setting, put the sharp point of the compass on point. Make a second large arc. It is important that the arcs intersect each other. Label the points of intersection of the arcs as and F. F (continued) U3-151
Use your straightedge to connect points and F. The point where F intersects is the midpoint of. Label this point G. G F 3. Put the sharp point of the compass on midpoint G and open the compass to point. Without changing the compass setting, draw an arc across the circle so it intersects the circle in two places. Label the points of intersection as H and J. H G J F U3-152
4. Use a straightedge to draw a line from point to point H and a second line from point to point J. H G J F o not erase any of your markings. H and F are both tangent to circle. U3-153
xample 4 ircle and circle are congruent. onstruct a line tangent to both circle and circle. 1. Use a straightedge to connect and, the centers of the circles. 2. t center point, construct a line perpendicular to. Label the point of intersection with circle as point. U3-154
3. t center point, construct a line perpendicular to. Label the point of intersection with circle as point. 4. Use a straightedge to connect points and. o not erase any of your markings. is tangent to circle and circle. U3-155