SAT
A tangent line is perpendicular to the radius at the point of tangency.
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Tangent and radius
A tangent line is perpendicular to the radius at the point of tangency.
Join the center to the point of tangency.
Negative reciprocal
Perpendicular slopes are negative reciprocals.
If the radius slope is ab, the tangent slope is −ba.
Example
The circle has center (3, 2). The line is tangent to the circle at (6, 6). What is the slope of the tangent line?
A tangent line is perpendicular to the radius at the point of tangency.
C is at (3, 2). P is at (6, 6).
The radius joins C to P.
Slope of the radius:
m = y₂ − y₁x₂ − x₁
m = 6 − 26 − 3
m = 43
Perpendicular slopes are negative reciprocals.
Slope of the tangent:
m = −34
Answer: −34
Work these three in a session.
The circle has center (2, 2). The line is tangent to the circle at (6, 5). What is the slope of the tangent line?
A tangent line is perpendicular to the radius at the point of tangency.
C is at (2, 2). P is at (6, 5).
The radius joins C to P.
Slope of the radius:
m = y₂ − y₁x₂ − x₁
m = 5 − 26 − 2
m = 34
Perpendicular slopes are negative reciprocals.
Slope of the tangent:
m = −43
34 is the slope of the radius.
−34 only changes the sign.
43 flips the fraction and keeps the sign.
Answer: A. −43
The circle has center (5, 5). The line is tangent to the circle at (2, 3). What is the slope of the tangent line?
A tangent line is perpendicular to the radius at the point of tangency.
C is at (5, 5). P is at (2, 3).
The radius joins C to P.
Slope of the radius:
m = y₂ − y₁x₂ − x₁
m = 3 − 52 − 5
m = −2−3
m = 23
Perpendicular slopes are negative reciprocals.
Slope of the tangent:
m = −32
23 is the slope of the radius.
−23 only changes the sign.
32 flips the fraction and keeps the sign.
Answer: A. −32
The circle has center (2, 5). The line is tangent to the circle at (5, 3). What is the slope of the tangent line?
A tangent line is perpendicular to the radius at the point of tangency.
C is at (2, 5). P is at (5, 3).
The radius joins C to P.
Slope of the radius:
m = y₂ − y₁x₂ − x₁
m = 3 − 55 − 2
m = −23
Perpendicular slopes are negative reciprocals.
Slope of the tangent:
m = 32
−23 is the slope of the radius.
23 only changes the sign.
−32 flips the fraction and keeps the sign.
Answer: D. 32
Try these five on your own.
The circle has center (1, 1). The line is tangent to the circle at (5, 3). What is the slope of the tangent line?
A tangent line is perpendicular to the radius at the point of tangency.
C is at (1, 1). P is at (5, 3).
The radius joins C to P.
Slope of the radius:
m = y₂ − y₁x₂ − x₁
m = 3 − 15 − 1
m = 24
m = 12
Perpendicular slopes are negative reciprocals.
Slope of the tangent:
m = −2
12 is the slope of the radius.
−12 only changes the sign.
2 flips the fraction and keeps the sign.
Answer: A. −2
The circle has center (2, 6). The line is tangent to the circle at (6, 4). What is the slope of the tangent line?
A tangent line is perpendicular to the radius at the point of tangency.
C is at (2, 6). P is at (6, 4).
The radius joins C to P.
Slope of the radius:
m = y₂ − y₁x₂ − x₁
m = 4 − 66 − 2
m = −24
m = −12
Perpendicular slopes are negative reciprocals.
Slope of the tangent:
m = 2
−12 is the slope of the radius.
12 only changes the sign.
−2 flips the fraction and keeps the sign.
Answer: D. 2
The circle has center (6, 8). The line is tangent to the circle at (4, 4). What is the slope of the tangent line?
A tangent line is perpendicular to the radius at the point of tangency.
C is at (6, 8). P is at (4, 4).
The radius joins C to P.
Slope of the radius:
m = y₂ − y₁x₂ − x₁
m = 4 − 84 − 6
m = −4−2
m = 2
Perpendicular slopes are negative reciprocals.
Slope of the tangent:
m = −12
2 is the slope of the radius.
−2 only changes the sign.
12 flips the fraction and keeps the sign.
Answer: B. −12
The circle has center (3, 5). The line is tangent to the circle at (4, 2). What is the slope of the tangent line?
A tangent line is perpendicular to the radius at the point of tangency.
C is at (3, 5). P is at (4, 2).
The radius joins C to P.
Slope of the radius:
m = y₂ − y₁x₂ − x₁
m = 2 − 54 − 3
m = −31
m = −3
Perpendicular slopes are negative reciprocals.
Slope of the tangent:
m = 13
−3 is the slope of the radius.
3 only changes the sign.
−13 flips the fraction and keeps the sign.
Answer: C. 13
The circle has center (6, 2). The line is tangent to the circle at (2, 5). What is the slope of the tangent line?
A tangent line is perpendicular to the radius at the point of tangency.
C is at (6, 2). P is at (2, 5).
The radius joins C to P.
Slope of the radius:
m = y₂ − y₁x₂ − x₁
m = 5 − 22 − 6
m = 3−4
m = −34
Perpendicular slopes are negative reciprocals.
Slope of the tangent:
m = 43
−34 is the slope of the radius.
34 only changes the sign.
−43 flips the fraction and keeps the sign.
Answer: D. 43
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