Wednesday, April 26, 2017 Warm Up

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Wednesday, April 26, 2017 Warm Up 1. What is the length of the diagonal of a square with side lengths ? 14 2. In a 45o45o90o triangle, what is the length of each leg if the hypotenuse is 8? 3. Find the value of x and y. x y 22 60o 4. In a 30o60o90o triangle, the shorter leg is 3.3 feet long. Find the perimeter. 5. Solve: 4x2 + 20 = 0

Properties of Tangents Wednesday, April 26, 2017 Essential Question: How do we use properties of a tangent to a circle? Lesson 6.1

Daily Homework Quiz 1. Give the name that best describes the figure . a. CD secant b. AB tangent c. FD d. EP Chord radius

Daily Homework Quiz 2. Tell how many common tangents the circles have . ANSWER One tangent; it is a vertical line through the point of tangency.

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.

Tangent Theorems Create right triangles for problem solving.

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

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

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

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

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

Write the binomial twice. Multiply. 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 Pythagorean Theorem (r + 50)2 = r2 + 802 Substitute. (r + 50)(r + 50) = r2 + 802 Write the binomial twice. Multiply. r2 + 50r +50r + 2500 = r2 + 6400 r2 + 100r + 2500 = r2 + 6400 Combine Like Terms. 100r = 3900 Subtract from each side. r = 39 ft . Divide each side by 100.

Subtract from each side. Find the radius of a circle 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. QT2 = QS2 + ST2 Pythagorean Theorem (r + 18)2 = r2 + 242 Substitute. r2 + 36r + 324 = r2 + 576 Multiply. 36r = 252 Subtract from each side. r = 7 Divide each side by 36.

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.

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

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

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

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