Lesson 6.1 – Properties of Tangent Lines to a Circle HW: Lesson 6.1/1-8

Using Properties of Tangents Radius to a Tangent Conjecture If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. D  𝐴𝐵 | 𝐵𝐷

Is TS tangent to R? Explain Using Properties of Tangents Is TS tangent to R? Explain If the Pythagorean Theorem works then the triangle is a right triangle  TS is tangent ? ? NO! ∆RST is not a right triangle so SR is not | to ST

In the diagram, AB is a radius of  A. Is BC tangent to  A? Explain. Using Properties of Tangents In the diagram, AB is a radius of  A. Is BC tangent to  A? Explain. If the Pythagorean Theorem works then the triangle is a right triangle  BC is tangent ? ? NO! ∆ABC is not a right triangle so AB is not | to BC

Using Properties of Tangents In the diagram, S is a point of tangency. Find the radius of r of circle T. 36+ r

In the diagram, is a radius of  P . Is  P tangent to ? Using Properties of Tangents In the diagram, is a radius of  P . Is  P tangent to ? If the Pythagorean Theorem works then the triangle is a right triangle  BC is tangent ? ? YES! ∆ABC is a right triangle so PT is | to TS

Using Properties of Tangents Tangent Segments Conjecture If two segments from the same exterior point are tangent to the circle, then they are congruent.

Using Properties of Tangents Tangent segments, from a common external point to their points of tangency, are congruent ● ●

Using Properties of Tangents x2 - 4 is tangent to R at S. is tangent to R at V. Find the value of x. Tangent segments are congruent

Using Properties of Tangents Any two tangent lines of a circle are equal in length. 2x + 10 = 3x + 7 2x + 3 = 3x 3 = x

Using Properties of Tangents In C, DA, is tangent at A and DB is tangent at B. Find x. ● ● 25= 6x -8 33= 6x 5.5 = x

Using Properties of Tangents PRACTICE Using Properties of Tangents

is tangent to  C at S and is tangent to C at T. Find the value of x is tangent to  C at S and is tangent to C at T. Find the value of x. is tangent to Q. Find the value of r. 28= 3x + 4 24= 3x 7 = x

Using Properties of Tangents A tangent line is perpendicular to the radius of a circle, therefore use the Pythagorean Theorem to solve for the unknown length. a2 = 62 + 82 a2 = 36 + 64 a2 = 100 a = 10

Using Properties of Tangents A tangent line is perpendicular to the radius of a circle, therefore use the Pythagorean Theorem to solve for the unknown length. Look for the length x, outside the circle. Let r be the radius of the circle, and let y = x + r. y2 = 122 + 162 y2 = 144 + 256 y2 = 400 y = 20 y Since y = x + r and r = 12 x + 12 = 20 x = 20 - 12 x = 8

Using Properties of Tangents x2 + 2 AB is tangent to C at B. AD is tangent to C at D. Find the value of x. AB = AD Two tangent segments from the same point are  11 = x2 + 2 Substitute values 9 = x2 Subtract 2 from each side. 3 = x Find the square root of 9.

Using Properties of Tangents Find the values of x, y, and z. All radii are ≅ y = 15 ∆UVR is a right triangle Tangent segments are ≅ z = 36

Using Properties of Tangents In the diagram, B is a point of tangency. Find the radius of  C

Using Properties of Tangents You are standing 14 feet from a water tower (R). The distance from you to a point of tangency (S) on the tower is 28 feet. What is the radius of the water tower? ● Radius = 21 feet Tower

Is tangent to C ?

Using Properties of Tangents Find the value of x.