MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

Warm UpWednesday, May 3, 2023

1. What is the length of the diagonal of a square with side lengths ?7 2

2. In a 45o45o90o triangle, what is the length of each leg if the hypotenuse is 8?

xy

2260o

3. Find the value of x and y.

4. In a 30o60o90o triangle, the shorter leg is 3.3 feet long. Find the perimeter.

5. Solve: 4x2 + 20 = 0

14

4 2

11, 11 3x y

9.9 3.3 3 feetP

5x i

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

Properties of Tangents

Essential Question:How do we use properties of a tangent to a circle?

Wednesday, May 3, 2023

Lesson 6.1

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

Daily Homework Quiz

1. Give the name that best describes the figure .

a. CD b. AB

c. FD d. EP

Chord

secant

radius

tangent

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

2. Tell how many common tangents the circles have .

ANSWER

One tangent; it is a vertical line through the point of tangency.

Daily Homework Quiz

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

Perpendicular Tangent Theorem 6.1

In a plane, if a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.

If l is tangent to Q at P, then l QP.

l

Q

P

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

Tangent Theorems

Create right triangles for problem solving.

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

1. Find the segment length indicated. Assume that lines which appear tangent are tangent.

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

2. Find the segment length indicated. Assume that lines which appear tangent are tangent.

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

3. Find the segment length indicated. Assume that lines which appear tangent are tangent.

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

4. Find the segment length indicated. Assume that lines which appear tangent are tangent.

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

5. Find the segment length indicated. Assume that lines which appear tangent are tangent.

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

Find the radius of a circle

6. In the diagram, ʘB is a point of tangency. Find the radius r of ʘC.

SOLUTION

You know 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 + 6400100r = 3900

r = 39 ft .

Pythagorean Theorem

Substitute.Write the binomial twice.

Subtract from each side.Divide each side by 100.

(r + 50)(r + 50) = r2 + 802

r2 + 50r +50r + 2500 = r2 + 6400Combine Like Terms.Multiply.

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

7. ST is tangent to ʘ Q. Find the value of r.

SOLUTION

You know from Theorem 10.1 that ST QS , so △ QST is a right triangle. You can use the Pythagorean Theorem.

r2 + 36r + 324 = r2 + 576

36r = 252

r = 7

Multiply.

Subtract from each side.

Divide each side by 36.

QT2 = QS2 + ST2

(r + 18)2 = r2 + 242

Pythagorean Theorem

Substitute.

Find the radius of a circle

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

Perpendicular Tangent Converse

In a plane, if a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle.

l

Q

P

If l QP at P, then l is tangent to Q.

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

8. Determine if line AB is tangent to the circle.

Verify a tangent to a circle

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

9. Determine if line AB is tangent to the circle. Verify a tangent to a circle

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

GUIDED PRACTICE

10. Is DE tangent to ʘ C?

ANSWER

Yes – The length of CE is 5 because the radius is 3 and the outside portion is 2. That makes ∆CDE a 3-4-5 Right Triangle. So DE and CD are

Verify a tangent to a circle

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

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 ʘ P at its endpoint on ʘ P. ST is tangent to ʘ P.

11. In the diagram, PT is a radius of ʘ P. Is ST tangent to ʘ P ?

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

HomeworkPage 186 # 12 – 17.Page 188 # 17 – 19.