In the second solution why isn't it a tangent of O when it intersects at one point C?

That's a good question, because point C is so clearly labeled, that it looks like that's the ONLY point of contact. But just because point C is labeled, doesn't mean it's the only point of contact, therefore it might not be a point of tangency, as we see from the angles. :-)

The first question answer is totally wired. If the reason of line *AC* to be tangent to O *circle* is because AC *line* perpendicular to OC *line*, then I can disprove that by shortening the length of OC *line*. _The first answer is more reasonable to me._

`OC` is a radius, to arbitrarily shorten it would completely alter the illustration. To test for perpendicularity, compute the unknown angle from the known angles: ``` 32 + 58 + x = 180 ``` if `x = 90º`, then you have conclusively proven perpendicularity.

How do you know that a line is a tangent?

It's *Tangent if… • it intersects at _only one point_ on the circumference, _AND_ • it _creates 90° angle with the radius*_, (therefore is perpendicular to the radius). Notice the *reference image is a "not to scale figure", _it only gives a semblance of the lines positions*_, so it is inaccurate, and *only used for visual cues to line arrangements*, not to indicate all the intersection points, not to estimate angles. So, we know both of *these lines*, (in these questions), *pass through a point on the circumference*, _but we can't tell if that's the only point by just looking at an imprecise reference picture_, so *check to see if a 90° is made between line and radius*. In these questions, _*only by doing the calculation can we know_ if the line meets the definition as Tangent*. ✯*Tangent* is to be both traits: • *single circumference intersect* • *perpendicular with radius*, (drawn from intersection point), therefore creates 90° angle ✯Never rely on a "Not to Scale Figure" to draw a conclusion, _always check by calculating the given values_. (ㆁωㆁ) Hope this helps.

What is an easy way to remember this if I am learning like three other things at a time

When you're learning several things at once, it can also help to connect the new ideas to others that you already have. Did you ever use regular polygons with lots of sides to approximate the area or the perimeter of a circle? If you did, you drew lots of tangent lines to the circle, and you figured out the measurements of the polygon by drawing right triangles using the radius as one side of the triangle and sections of a tangent line as another side. So remembering that picture can help us remember that the line tangent to a circle at a point is at a right angle to the radius that touches the same point. You could also try watching the video _Proof: Radius is perpendicular to tangent line_. It shows you a way of convincing yourself that the tangent line must be perpendicular to the radius.

In the second solution why isn't it a tangent of O when it intersects at one point C?

Add up the two other interior angles of triangle ABC. The sum of Angle A (51 degrees) and Angle B (49 degrees) is 100 degrees, so measure of angle ACB is actually 80 degrees, even though it appears to be a right angle. Therefore, segment AC is not perpendicular to segment BC.

If right triangle are other 2 angles 45 degrees

Not necessarily. The other two angles could be any numbers as long as they add up to 90 degrees.

Main content

Course: High school geometry > Unit 8

Lesson 10: Properties of tangents

Determining tangent lines: angles

Google Classroom

Solve two problems that apply properties of tangents to determine if a line is tangent to a circle.

Problem 1

Segment

\overset{―}{O C}

is a radius of circle

O

Note: Figure not necessarily drawn to scale.

Is line $\overset{\leftrightarrow}{A C}$ ‍ tangent to circle $O$ ‍?

(Choice A)
Yes, because $\overset{―}{A C}$ ‍ intersects circle $O$ ‍ at point $C$ ‍
(Choice B)
Yes, because $\overset{―}{A C}$ ‍ is perpendicular to $\overset{―}{O C}$ ‍
(Choice C)
No, because $\overset{―}{A C}$ ‍ is not perpendicular to $\overset{―}{O C}$ ‍
(Choice D)
No, because $△ A O C$ ‍ intersects circle $O$ ‍ at more than one point

line

that is tangent to a circle at a particular point is perpendicular to the

radius

at that point.

The interior angles of a triangle sum to

180^{\circ}

. Let's check whether

∠ A C O

is a right angle.

$\begin{aligned} m ∠ C A O + m ∠ A O C + m ∠ A C O & = 180^{\circ} \\ 32^{\circ} + 58^{\circ} + m ∠ A C O & = 180^{\circ} \\ m ∠ A C O & = 90^{\circ} \end{aligned}$ ‍

Angle

∠ A C O

is a right angle.

Yes, $\overset{\leftrightarrow}{A C}$ ‍ is tangent to circle $O$ ‍, because $\overset{―}{A C}$ ‍ is perpendicular to $\overset{―}{O C}$ ‍.

Problem 2

Segment

\overset{―}{B C}

is a diameter of circle

O

Note: Figure not necessarily drawn to scale.

Is line $\overset{\leftrightarrow}{A C}$ ‍ tangent to circle $O$ ‍?

(Choice A)
Yes, because $\overset{―}{A C}$ ‍ intersects circle $O$ ‍ at point $C$ ‍
(Choice B)
Yes, because $\overset{―}{A C}$ ‍ is perpendicular to $\overset{―}{B C}$ ‍
(Choice C)
No, because $\overset{―}{A C}$ ‍ is not perpendicular to $\overset{―}{B C}$ ‍
(Choice D)
No, because $△ A B C$ ‍ intersects circle $O$ ‍ at more than one point

line

that is tangent to a circle at a particular point is perpendicular to the

radius

at that point. Thus, it is also perpendicular to the diameter at the same point.

The interior angles of a triangle sum to

180^{\circ}

. Let's check whether

∠ A C B

is a right angle.

$\begin{aligned} m ∠ B A C + m ∠ A B C + m ∠ A C B & = 180^{\circ} \\ 51^{\circ} + 49^{\circ} + m ∠ A C B & = 180^{\circ} \\ m ∠ A C B & = 80^{\circ} \end{aligned}$ ‍

Angle

∠ A C B

is not a right angle.

No, $\overset{\leftrightarrow}{A C}$ ‍ is not tangent to circle $O$ ‍, because $\overset{―}{A C}$ ‍ is not perpendicular to $\overset{―}{B C}$ ‍.

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Jeiel Damina
Posted 8 years ago. Direct link to Jeiel Damina's post “In the second solution wh...”
In the second solution why isn't it a tangent of O when it intersects at one point C?
Button navigates to signup pageComment on Jeiel Damina's post “In the second solution wh...”
(34 votes)
Answer
- Charles Breiling
  Posted 6 years ago. Direct link to Charles Breiling's post “That's a good question, b...”
  That's a good question, because point C is so clearly labeled, that it looks like that's the ONLY point of contact. But just because point C is labeled, doesn't mean it's the only point of contact, therefore it might not be a point of tangency, as we see from the angles. :-)
  Comment on Charles Breiling's post “That's a good question, b...”
  (16 votes)
RandomDad
Posted 8 years ago. Direct link to RandomDad's post “The first question answer...”
The first question answer is totally wired. If the reason of line AC to be tangent to O circle is because AC line perpendicular to OC line, then I can disprove that by shortening the length of OC line.
The first answer is more reasonable to me.

Button navigates to signup pageComment on RandomDad's post “The first question answer...”
(8 votes)
Answer
- redthumb.liberty
  Posted 8 years ago. Direct link to redthumb.liberty's post “`OC` is a radius, to arbi...”
  OC is a radius, to arbitrarily shorten it would completely alter the illustration. To test for perpendicularity, compute the unknown angle from the known angles:
  
  32 + 58 + x = 180
  
  if x = 90º, then you have conclusively proven perpendicularity.
  Comment on redthumb.liberty's post “`OC` is a radius, to arbi...”
  (12 votes)
Samantha Ann
Posted 8 years ago. Direct link to Samantha Ann's post “If right triangle are oth...”
If right triangle are other 2 angles 45 degrees
Button navigates to signup pageButton navigates to signup page
(0 votes)
Answer
- Sola Fatade
  Posted 8 years ago. Direct link to Sola Fatade's post “Not necessarily. The othe...”
  Not necessarily. The other two angles could be any numbers as long as they add up to 90 degrees.
  Comment on Sola Fatade's post “Not necessarily. The othe...”
  (20 votes)
jayden.choong
Posted 3 years ago. Direct link to jayden.choong's post “How do you know that a li...”
How do you know that a line is a tangent?
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(2 votes)
Answer
- Seed Something
  Posted 2 years ago. Direct link to Seed Something's post “It's *Tangent if… • it in...”
  It's Tangent if…
  • it intersects at only one point on the circumference,
  AND
  • it creates 90° angle with the radius, (therefore is perpendicular to the radius).
  
  Notice the reference image is a "not to scale figure", it only gives a semblance of the lines positions, so it is inaccurate, and only used for visual cues to line arrangements, not to indicate all the intersection points, not to estimate angles.
  
  So, we know both of these lines, (in these questions), pass through a point on the circumference, but we can't tell if that's the only point by just looking at an imprecise reference picture, so check to see if a 90° is made between line and radius.
  
  In these questions, only by doing the calculation can we know if the line meets the definition as Tangent.
  
  ✯Tangent is to be both traits:
  • single circumference intersect
  • perpendicular with radius, (drawn from intersection point), therefore creates 90° angle
  
  ✯Never rely on a "Not to Scale Figure" to draw a conclusion, always check by calculating the given values.
  
  (ㆁωㆁ) Hope this helps.
  Button navigates to signup page
  (17 votes)
346028
Posted 5 years ago. Direct link to 346028's post “What is an easy way to re...”
What is an easy way to remember this if I am learning like three other things at a time
Button navigates to signup pageButton navigates to signup page
(2 votes)
Answer
- Charlie Auen
  Posted 4 years ago. Direct link to Charlie Auen's post “When you're learning seve...”
  When you're learning several things at once, it can also help to connect the new ideas to others that you already have. Did you ever use regular polygons with lots of sides to approximate the area or the perimeter of a circle? If you did, you drew lots of tangent lines to the circle, and you figured out the measurements of the polygon by drawing right triangles using the radius as one side of the triangle and sections of a tangent line as another side. So remembering that picture can help us remember that the line tangent to a circle at a point is at a right angle to the radius that touches the same point.
  You could also try watching the video Proof:
  Radius is perpendicular to tangent line. It shows you a way of convincing yourself that the tangent line must be perpendicular to the radius.
  Button navigates to signup page
  (7 votes)
Mary
Posted 3 years ago. Direct link to Mary's post “If a line is tangent to a...”
If a line is tangent to a circle, does that mean that the point of intersection between the line and the circle is also a right angle?
Button navigates to signup pageComment on Mary's post “If a line is tangent to a...”
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JawanE
Posted a year ago. Direct link to JawanE's post “In the second solution wh...”
In the second solution why isn't it a tangent of O when it intersects at one point C?
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(2 votes)
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- LillaS
  Posted a year ago. Direct link to LillaS's post “Add up the two other inte...”
  Add up the two other interior angles of triangle ABC. The sum of Angle A (51 degrees) and Angle B (49 degrees) is 100 degrees, so measure of angle ACB is actually 80 degrees, even though it appears to be a right angle. Therefore, segment AC is not perpendicular to segment BC.
  Button navigates to signup page
  (5 votes)
rickela.joseph
Posted 8 years ago. Direct link to rickela.joseph's post “how do you now that trian...”
how do you now that triangles in a circle are alternate if no sides where given ?
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am11
Posted 4 years ago. Direct link to am11's post “Do opposite angles always...”
Do opposite angles always add up to 180 degrees?
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- Jordan Rivalet
  Posted 3 years ago. Direct link to Jordan Rivalet's post “Yes, for a quadrilateral ...”
  Yes, for a quadrilateral inscribed within a circle.
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Taliyah
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I have a pretest on lines and angles and I can't find nothing to help me understand it, I am so confused.
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