properties of tangents. example 1 identify special segments and lines tell whether the line, ray, or...
TRANSCRIPT
Properties Properties of of
TangentsTangents
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.
Theorem 6.1
In a plane, a line segment is tangent to a circle if and only if the line is perpendicular to a radius of
the circle at its endpoint on the circle.
EXAMPLE 3 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 6.1, ST is tangent to P.
In the diagram, PT is a radius of P. Is ST tangent to P ?
EXAMPLE 4 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 6.1 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 6.2
Tangent segments from a common external point are
congruent.
EXAMPLE 5
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
Is DE tangent to C?
ANSWER
Yes
GUIDED PRACTICE
ST is tangent to Q.Find the value of r.
ANSWER
r = 7
GUIDED PRACTICE
Find the value(s) of x.
+3 = x
ANSWER