Lesson 6.1 – Properties of Tangent Lines to a Circle

HW: Lesson 6.1/1-8

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

Radius to a Tangent Conjecture

Using Properties of Tangents

D

|

45

43

11

R

S T

Is TS tangent to R? ExplainIf the Pythagorean Theorem works then the triangle is a right triangle TS is tangent

222 114345 ? 12118492025

?

19702025

NO! ∆RST is not a right triangle so SR is not | to ST

Using Properties of Tangents

In the diagram, AB is a radius of A.

Is BC tangent to A? Explain.

Using Properties of Tangents

If the Pythagorean Theorem works then the triangle is a right triangle BC is tangent

222 602567 ? 36006254489

?

42254489

NO! ∆ABC is not a right triangle so AB is not | to BC

In the diagram, S is a point of tangency. Find the radius of r of circle T.

Using Properties of Tangents

222 4836 rr 22 2304721296 rrr

2304721296 r

100872 r

14r

36+ r

In the diagram, is a radius of P . Is P tangent to ?

PT

ST

Using Properties of Tangents

If the Pythagorean Theorem works then the triangle is a right triangle BC is tangent

222 123537 ? 14412251369

?

13691369

YES! ∆ABC is a right triangle so PT is | to TS

If two segments from the same exterior point are tangent to the circle, then they are congruent.

Tangent Segments Conjecture

ACAB

Using Properties of Tangents

Tangent segments, from a common external point to their points of tangency, are congruent

Using Properties of Tangents

●

●

21

R

S

U

V

x2 - 4US is tangent to R at S. is tangent to R at V. Find the value of x.

UV

2142 x

252 x5x

Tangent segments are congruent

Using Properties of Tangents

Any two tangent lines of a circle are equal in length.

2x + 10 = 3x + 72x + 3 = 3x

3 = x

Using Properties of Tangents

In C, DA, is tangent at A and DB is tangent at B. Find x.

Using Properties of Tangents

●

●

25= 6x -833= 6x5.5 = x

PRACTICEUsing Properties of Tangents

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.

RS RT

ST

28= 3x + 424= 3x7 = x

22 57636324 rrr

222 2418 rr

57636324 r

25236 r7r

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 + 64a2 = 100a = 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 + 256y2 = 400y = 20

x + 12 = 20x = 20 - 12

x = 8

Since y = x + r and r = 12y

Using Properties of Tangents

AB is tangent to C at B.

AD is tangent to C at D.

Find the value of x. 11

AC

B

Dx2 + 2

11 = x2 + 2

Two tangent segments from the same point are

Substitute values

AB = AD

9 = x2 Subtract 2 from each side.

3 = x Find the square root of 9.

Using Properties of Tangents

x

z 15

y36

R

S

U

V

Find the values of x, y, and z.

All radii are ≅y = 15

222 1536 x22512962 x

15212 x39x

Tangent segments are ≅z = 36

∆UVR is a right triangle

Using Properties of Tangents

In the diagram, B is a point of tangency. Find the radius of C

Using Properties of Tangents

222 8050 rr 22 64001002500 rrr

64001002500 r

3900100 r39r

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?

r

14 ft

28 ft

r

R

S T

Radius = 21 feetTower

222 2814 rr

222 2819628 rrr

78419628 r58828 r

21r

●

Using Properties of Tangents

Is tangent to C ?DE

Find the value of x.

Using Properties of Tangents