lesson 6.1 – properties of tangent lines to a circle
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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. . |. Is TS tangent to R ? Explain. - PowerPoint PPT PresentationTRANSCRIPT
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