Circles - Tangent Lines and Theorems

Tangent Line Properties

Theorem works both ways:
- If line AB is tangent to the circle at point P, then it's perpendicular to the radius.
- If a line is perpendicular to the radius, then it's a tangent.

Two angles formed by the tangent and radii at the point of tangency are congruent.
These angles measure 90 degrees.
Given three angles in a quadrilateral, the missing angle can be found by setting the sum equal to 360 degrees.
- Example: x + 90 + 135 + 90 = 360

To check if a line (MP) is tangent to a circle with center N, verify if the angle NMP is a right angle.
Check if the sum of angles in triangle NMP equals 180 degrees.
- 58 + 33 + \angle NMP = 180
If the calculated angle NMP is not 90 degrees, then line MP is not tangent.
Example Calculation:
- 180 - 58 - 33 = 89
- Since 89 \neq 90, line MP is not tangent.

If segment LK is tangent to circle N at point L, then angle NLK is a right angle.
This implies dealing with a right triangle, allowing the use of the Pythagorean theorem.
Using the Pythagorean theorem: c^2 = a^2 + b^2
- c is the hypotenuse (opposite the right angle).
- a and b are the other two sides.
Identify the lengths:
- a = 9 (radius)
- b = x + 5
- c = 9 + x - 1 = x + 8
Setup the equation:
- (x+8)^2 = 9^2 + (x+5)^2
Expand and simplify:
- x^2 + 16x + 64 = 81 + x^2 + 10x + 25
Combine like terms:
- x^2 + 16x + 64 = x^2 + 10x + 106
- 6x = 42
- x = 7
Remember to answer the original question. If they ask for KN (where KN = x - 1 + 9), then:
- KN = 7 - 1 + 9 = 15

Given a line equation 3x - 4y = 19 tangent to a circle centered at (2, 3), find the tangent point.
Convert the line equation to slope-intercept form (y = mx + b).
- -4y = -3x + 19
- y = \frac{3}{4}x - \frac{19}{4}
Find the slope of the line perpendicular to it (the radius):
- The slope will be the opposite reciprocal: m_{\perp} = -\frac{4}{3}
Use the point-slope form to find the equation of the line passing through the center (2, 3) with the perpendicular slope.
y - y1 = m(x - x1)
y - 3 = -\frac{4}{3}(x - 2)
y = -\frac{4}{3}x + \frac{8}{3} + 3
y = -\frac{4}{3}x + \frac{17}{3}
Convert this to standard form: 4x + 3y = 17

Solve the system of equations formed by the tangent line and the line containing the radius to find the intersection point (tangent point).
Use elimination method:
- 3x - 4y = 19
- 4x + 3y = 17
Multiply equations to eliminate a variable:
- Multiply the first equation by 3 and the second by 4:
  - 9x - 12y = 57
  - 16x + 12y = 68
Add the equations to eliminate y:
- 25x = 125
- x = 5
Substitute x = 5 back into either equation to find y:
- Using 4x + 3y = 17:
  - 4(5) + 3y = 17
  - 20 + 3y = 17
  - 3y = -3
  - y = -1
The tangent point is (5, -1).

Given a circle and a tangent line, find the equation of a second tangent line with the same slope.
The two tangent lines are parallel.
Use a point on the second tangent line and the slope.
If the second point of tangency is at (−1,7), then using slope 3/4:
y - 7 = \frac{3}{4}(x + 1)
y = \frac{3}{4}x + \frac{3}{4} + 7
y = \frac{3}{4}x + \frac{31}{4}

Given two tangent lines from the same external point to a circle, the triangles formed by radii to the tangent points are congruent.
These tangent lines are perpendicular to the radii, forming right angles.
Hypotenuse Leg (HL) Theorem: If the hypotenuse and a leg of one right triangle are congruent to the corresponding parts of another right triangle, the triangles are congruent.
Corresponding Parts of Congruent Triangles are Congruent (CPCTC):
- If \[\triangle OYZ \cong \triangle OXZ], then segments YZ = XZ.
- Angle bisector, e.g. zt is an angle bisector because \[\angle TXZ = \angle TYZ]

If TX = 12 and TZ = 20, use triangles to derive length of XZ/ZY with Pythagorean theorem.
\sqrt{20^2 - 12^2} = 16
XZ = ZY = 16

If two segments with a common endpoint exterior to the circle are tangent to the circle, then the segments are congruent.
If AB and AC are tangent to the circle from point A, then AB = AC.
Tangent segments from the same external point are always congruent.

Satellite requires line of sight for communication with ground stations; ground stations are points of tangency.
Earth's radius: 6,371 km.
Satellite altitude: 35,786 km above Earth.
Problem: Find the amount of time needed for a signal to travel from one station to the satellite and then to the other station.

Use the Pythagorean Theorem to find this distance.
Total Length from the earth's center to the satellite is \[6371 + 35786 = 42157km \]
Segment from the ground station = \[ (42157)^2 - (6371)^2 = 41,673 km\]
Once we calculate the length, we need to calculate the amount of time needed for a signal to go from one station up to the satellite and down to the other station.
Use the formula: distance = speed \times time
Signal speed: 300,000 km/second.
41673 = 300000 \times t
t \approx 0.14 seconds

Given a quadrilateral ABCD circumscribed around a circle, find its perimeter.
Utilize that tangent segments from the same point are congruent. From that, we can determine values like AD, BC, etc.
Add all side lengths to find the perimeter.