Tangents and circles

1
Tangents and circles

• A tangent lies in the same plane as a circle and
  intersects the circle at exactly one point
• A radius drawn to a point of tangency meets the
  tangent at a fixed angle, which is 90 degrees

2
Theorem 58-1

• If a line is drawn to a circle, then the line is
  perpendicular to a radius drawn to the point of
  tangency.
• A point of tangency is a point where a tangent
  intersects a circle.

3
Theorem 58-2

• If a line in the plane of a circle is
  perpendicular to the radius at its endpoint on
  the circle, then the line is tangent to the
  circle (this is the converse of theorem 58-1)

4
??????????????

• If 2 tangents to the same circle intersect, the
  tangent segments have a special property.
• What is it?

5
Theorem 58-3

• If 2 tangents are drawn to a circle from the same
  exterior point, then they are congruent- CE is
  congruent to DE

6
????????????????????

• If tangent lines are drawn to the endpoints of a
  diameter of a circle, where will they intersect?
• They will not intersect, they will be parallel

7
Applying relationships of tangents from an
exterior point

• Determine the perimeter of ACED

17 in.
8 in
8
Practice

• Line a is tangent to circle R at D, and line b
  passes through R. Lines a and b intersect at E.
  If mltRED 42o,determine mltDRE

9
practice

• Let CA be a radius of circle C. Let line m be a
  tangent to circle C at A. Let B be an exterior
  point of circle C, with mlt BAClt90o. Is line AB a
  tangent to circle C?
• Why or why not?

10
practice

• Circle C has a radius of 5 inches. Line segments
  XZ and YZ are tangents to circle C and Z is
  exterior to circle C. If ltXCY is aright angle,
  what is the area of quad. CXZY?

11
practice

• A decorative window is shaped like a triangle
  with an inscribed circle. If the triangle is an
  equilateral triangle, the circle has a radius of
  3 ft., and DC is 6 ft., what is the perimeter of
  the triangle in simplified radical form.

D
12
Lab 8- Tangents to a circle through a point on
the circle

• 1. Draw circle A, and a point on the circle B
• 2. Draw AB
• 3. Construct the line perpendicular to AB through
  B( construction lab 2)- the perpendicular line is
  tangent to circle A at B

13
Construct lines tangent to a circle from a point
not on the circle

• 1. Draw a circle and label the center C.
• 2. Choose a point exterior to the circle, and
  label it P. Draw CP.
• 3.Costruct the midpoint of CP. Label the midpoint
  M.
• 4. Draw a circle with a radius CM, centered on M.
  (Notice that P is also on this circle)
• 5. Label the points of intersection X and Y.
• Draw XP. This line is tangent to circle C at
  point X. Draw YP. Notice that YP is tangent to
  circle C at point Y.

14
Construction practice

• Construct (points A is on circle D and points F
  and G are outside , but near to circle D and E)
• 1. a line tangent to circle D at point A
• 2. a line tangent to circle D from point F
• 3. a line tangent to circle E from point F
• 4. a line tangent to circle E from point G

15
Practice

• Let CA be a radius of circle C. Let line m be
  tangent to circle C at A. Let B be an exterior
  point of circle C, with mltBAC lt90o. Is AB a
  tangent to circle C? Why or why not?

16
Practice

• Circle c has a 5-inch radius. XZ and YZ are
  tangents to circle C and Z is exterior to circle
  C. If ltXYC is a right angle, what is the area of
  quad. CXZY?

17
Practice

• A decorative window is shaped like a triangle
  with an inscribed circle. If the triangle is an
  equilateral triangle, the circle has a radius of
  3 feet, and CQ is 6 feet, what is the perimeter
  of the triangle in simplified radical form?

Q
18
Lab 8

• Construct lines tangent to a circle through a
  point on the circle
• 1. begin with circle A and a point on the circle,
  B
• 2. Draw AB
• 3. construct the line perpendicular to AB through
  B
• This line is tangent to circle A at B

19
Construct lines tangent to a circle through a
point not on the circle

• 1. Draw a circle and label the center C
• 2. Choose a point exterior to the circle, and
  label it P.
• 3. draw CP
• 4. Construct the midpoint of CP. Label the
  midpoint M
• 5. Draw a circle with radius, CM centered on M.
  (notice that P is also on this circle)
• 6. Label the points of intersection of circle C
  and circle M as points X and Y
• 7.Draw XP. This line is tangent to circle C at
  point x. Draw YP. This line is tangent to circle
  C at point Y

20
Practice

• Use a compass to draw circle D and circle E. Draw
  point A on circle D. Draw points F and G outside,
  but near to circle d and circle E.
• Construct
• A) a line tangent to circle D at point A
• B) a line tangent to circle D from point F
• C) a line tangent to circle E from point F
• D) a line tangent to circle E from point G