tangency. lines of circles example 1 identify special segments and lines tell whether the line, ray,...
TRANSCRIPT
Tangency
Lines of Circles
EXAMPLE 1 Identify special segments and lines
Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of C.
ACa.
SOLUTION
is a radius because C is the center and A is a point on the circle.
ACa.
EXAMPLE 1 Identify special segments and lines
b. AB is a diameter because it is a chord that contains the center C.
Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of C.
b. AB
SOLUTION
EXAMPLE 1 Identify special segments and lines
c. DE is a tangent ray because it is contained in a line that intersects the circle at only one point.
Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of C.
SOLUTION
DEc.
EXAMPLE 1 Identify special segments and lines
d. AE is a secant because it is a line that intersects the circle in two points.
Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of C.
SOLUTION
AEd.
SOLUTION
GUIDED PRACTICE for Example 1
Is a chord because it is a segment whose endpoints are on the circle.
AG
CB is a radius because C is the center and B is a point on the circle.
1. In Example 1, what word best describesAG ? CB ?
SOLUTION
GUIDED PRACTICE for Example 1
2. In Example 1, name a tangent and a tangent segment.
A tangent is DE
A tangent segment is DB
EXAMPLE 2 Find lengths in circles in a coordinate plane
b. Diameter of A
Radius of Bc.
Diameter of Bd.
Use the diagram to find the given lengths.
a. Radius of A
SOLUTION
a. The radius of A is 3 units.
b. The diameter of A is 6 units.
c. The radius of B is 2 units.
d. The diameter of B is 4 units.
SOLUTION
GUIDED PRACTICE for Example 2
a. The radius of C is 3 units.
b. The diameter of C is 6 units.
c. The radius of D is 2 units.
d. The diameter of D is 4 units.
3. Use the diagram in Example 2 to find the radius and diameter of C and D.
Diagram Common tangents
EXAMPLE 3 Draw common tangents
Tell how many common tangents the circles have and draw them.
a. b. c.
SOLUTION
a. 4 common tangents 3 common tangentsb.
EXAMPLE 3 Draw common tangents
c. 2 common tangents
Tell how many common tangents the circles have and draw them.
c.
SOLUTION
SOLUTION
GUIDED PRACTICE for Example 3
Tell how many common tangents the circles have and draw them.4.
2 common tangents
SOLUTION
GUIDED PRACTICE for Example 3
Tell how many common tangents the circles have and draw them.
1 common tangent
5.
SOLUTION
GUIDED PRACTICE for Example 3
Tell how many common tangents the circles have and draw them.
No common tangents
6.
Theorem Tangent Perpendicular
Theorem Tangent Perpendicular Converse
EXAMPLE 4 Verify a tangent to a circle
SOLUTION
Use the Converse of the Pythagorean Theorem. Because 122 + 352 = 372, PST is a right triangle and ST PT . So, ST is perpendicular to a radius of P at its endpoint on P. By Theorem 10.1, ST is tangent to P.
In the diagram, PT is a radius of P. Is ST tangent to P ?
EXAMPLE 5 Find the radius of a circle
In the diagram, B is a point of tangency. Find the radius r of C.
SOLUTION
You know from Theorem 10.1 that AB BC , so ABC is a right triangle. You can use the Pythagorean Theorem.
AC2 = BC2 + AB2
(r + 50)2 = r2 + 802
r2 + 100r + 2500 = r2 + 6400
100r = 3900
r = 39 ft .
Pythagorean Theorem
Substitute.
Multiply.
Subtract from each side.
Divide each side by 100.
Theorem Common Point Tangency
EXAMPLE 6 Find the radius of a circle
RS is tangent to C at S and RT is tangent to C at T. Find the value of x.
SOLUTION
RS = RT
28 = 3x + 4
8 = x
Substitute.
Solve for x.
Tangent segments from the same point are
GUIDED PRACTICE for Examples 4, 5 and 6
7. Is DE tangent to C?
ANSWER
Yes
GUIDED PRACTICE for Examples 4, 5 and 6
8. ST is tangent to Q.Find the value of r.
ANSWER
r = 7
GUIDED PRACTICE for Examples 4, 5 and 6
9. Find the value(s) of x.
+3 = x
ANSWER