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10.1
Warm UpWarm Up
Lesson QuizLesson Quiz
Lesson PresentationLesson Presentation
Use Properties of Tangents
10.1 Warm-Up
1. What measure is needed to find the circumferenceor area of a circle?
2. Find the radius of a circle with diameter 8 centimeters.
ANSWER radius or diameter
ANSWER 4 cm
3. A right triangle has legs with lengths 5 inches and12 inches. Find the length of the hypotenuse.
ANSWER 13 in.
10.1 Warm-Up
4. Solve 6x + 15 = 33.
ANSWER 3
5. Solve (x + 18)2 = x2 + 242.
ANSWER 7
10.1 Example 1
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.
b. AB
b. AB is a diameter because it is a chord that contains the center C.
10.1 Example 1
Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of C.
SOLUTION
c. DE is a tangent ray because it is contained in a line that intersects the circle at only one point.
DEc.
d. AE is a secant because it is a line that intersects the circle in two points.
AEd.
10.1 Guided Practice
1. In Example 1, what word best describes AG? CB?
ANSWER chord; radius
2. In Example 1, name a tangent and a tangent segment.
ANSWER DE, DB
10.1 Example 2
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.
10.1 Guided Practice
3. Use the diagram in Example 2 to find the radius and diameter of C and D.
ANSWER 3; 6; 2; 4
10.1 Example 3
Tell how many common tangents the circles have and draw them.
a.
SOLUTION
a. 4 common tangents
10.1 Example 3
Tell how many common tangents the circles have and draw them.
SOLUTION
b.
3 common tangentsb.
10.1 Example 3
Tell how many common tangents the circles have and draw them.
SOLUTION
c.
c. 2 common tangents
10.1 Guided Practice
Tell how many common tangents the circles have and draw them.
4.
2 common tangents
ANSWER
10.1 Guided Practice
Tell how many common tangents the circles have and draw them.
5.
1 common tangent
ANSWER
10.1 Guided Practice
Tell how many common tangents the circles have and draw them.
no common tangents
ANSWER
6.
10.1 Example 4
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 ?
10.1 Example 5
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.
10.1 Example 6
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 .
10.1 Guided Practice
7. Is DE tangent to C?
ANSWER Yes
10.1 Guided Practice
ANSWER 7
8. ST is tangent to Q.Find the value of r.
10.1 Guided Practice
ANSWER ±3
9. Find the value(s) of x.
10.1 Lesson Quiz
ANSWER secant
1. Give the name that best describes the figure .
a. CD b. AB
c. FD d. EP
ANSWER tangent
ANSWER chord ANSWER radius
10.1 Lesson Quiz
ANSWER
One tangent; it is a vertical line through the point of tangency.
2. Tell how many common tangents the circles have .
10.1 Lesson Quiz
3. Is AB tangent to C? Explain. .
ANSWER
Yes; 16 + 30 = 1156 = 34 so AB AC, and a line to a radius at its endpoint is tangent to the circle.
2 22
10.1 Lesson Quiz
4. Find x.
ANSWER 12
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