Unit 6, Lesson 3.1 Constructing Tangent Lines

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1 Unit 6, Lesson 3.1 Constructing Tangent Lines 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. 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. Exactly one tangent line can be constructed by using construction tools to create a line perpendicular to the radius at a point on the circle. Constructing a Tangent at a Point on a Circle Using a Compass 1. Use a straightedge to draw a ray from center O through the given point P on the circle. e sure the ray extends past point P. 2. Construct the line perpendicular to OPat point P. This is the same procedure as constructing a perpendicular line through 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. Draw 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 on one side of OP. d. Without changing your compass setting, put the sharp point of the compass on. Make a second large arc on the same side of OP. It is important that the arcs intersect each other. Label this point of intersection C. 3. Use your straightedge to connect the point C of intersection of the arcs and point P. 4. Label the new line m. 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. Exactly two lines can be constructed that are tangent to the circle through an exterior point not on the circle. The two segments from the external point to each point of tangency are congruent. Unit6Lesson /21/2018

2 Constructing a Tangent from an Exterior Point Not on a Circle Using a Compass 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 C and D. d. Use your straightedge to connect points C and D. e. The point where CD 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. 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. 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. Common 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. Unit6Lesson /21/2018

3 Common Errors/Misconceptions assuming that a radius and a line are perpendicular at the possible point of intersection simply by observation assuming two tangent segments are congruent by observation incorrectly changing the compass settings not making large enough arcs to find the points of intersection Example 1 Use a compass and a straightedge to construct C tangent to circle at point. 1. Draw a ray from center through point and extending beyond point. 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 D and E. Unit6Lesson /21/2018

4 Example 1 (continued) 3. Put the sharp point of the compass on point D and set it to a width greater than the distance of D. Make a large arc intersecting. 4. Without changing the compass setting, put the sharp point of the compass on point E and draw a second arc that intersects the first. Label the point of intersection with the arc drawn in step 3 as point C. 5. Draw a line connecting points C and, creating tangent C. C is tangent to circle at point. Unit6Lesson /21/2018

5 Example 2 Using the circle and tangent line from Example 1, construct two additional tangent lines, so that circle below will be inscribed in a triangle. 1. Choose a point, G, on circle. G 2. Draw a ray from center to point G. G Unit6Lesson /21/2018

6 Example 2 (continued) 3. Follow the process explained in Example 1 for constructing a tangent line through point G. G 4. Choose another point, H, on circle. Draw a ray from center to point H, and follow the process explained in Example 1 to construct the third tangent line. e sure to draw the tangent lines long enough to intersect one another. H G Circle is inscribed in a triangle. Unit6Lesson /21/2018

7 Example 3 Use a compass and a straightedge to construct the lines tangent to circle C at point D. 1. Draw a ray connecting center C and the given point D. 2. Find the midpoint of CDby constructing the perpendicular bisector. Put the sharp point of your compass on point C. Open the compass wider than half the distance of CD. Make a large arc intersecting CD. Without changing your compass setting, put the sharp point of the compass on point D. Make a second large arc. It is important that the arcs intersect each other. Label the points of intersection of the arcs as E and F. Unit6Lesson /21/2018

8 Example 3 (continued) Use your straightedge to connect points E and F. The point where EF intersects CD is the midpoint of CD. Label this point G. 3. Put the sharp point of the compass on midpoint G and open the compass to point C. 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. 4. Use a straightedge to draw a line from point D to point H and a second line from point D to point J. DH and DF are both tangent to circle C. Unit6Lesson /21/2018

9 Example 4 Circle and circle are congruent. Construct 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 D. D Unit6Lesson /21/2018

10 Example 4 (continued) 3. t center point, construct a line perpendicular to. Label the point of intersection with circle as point E. D E 4. Use a straightedge to connect points D and E. D E DE is tangent to circle and circle. Unit6Lesson /21/2018