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# Determining tangent lines: angles

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

## Problem 1

Segment $\stackrel{―}{OC}$ is a radius of circle $O$.
Note: Figure not necessarily drawn to scale.
Is line $\stackrel{↔}{AC}$ tangent to circle $O$?

## Problem 2

Segment $\stackrel{―}{BC}$ is a diameter of circle $O$.
Note: Figure not necessarily drawn to scale.
Is line $\stackrel{↔}{AC}$ tangent to circle $O$?

## Want to join the conversation?

• 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.
• 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.
• 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.
• 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?
• 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.