Main content

### Course: High school geometry > Unit 8

Lesson 10: Properties of tangents- Proof: Radius is perpendicular to tangent line
- Determining tangent lines: angles
- Determining tangent lines: lengths
- Proof: Segments tangent to circle from outside point are congruent
- Tangents of circles problem (example 1)
- Tangents of circles problem (example 2)
- Tangents of circles problem (example 3)
- Tangents of circles problems
- Challenge problems: radius & tangent
- Challenge problems: circumscribing shapes

© 2024 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Tangents of circles problem (example 2)

Sal finds a missing angle using the property that tangents are perpendicular to the radius. Created by Sal Khan.

## Want to join the conversation?

- At1:35, why is a line that is tangent to a circle perpendicular to the radius of that circle?(26 votes)
- It is because any line that is tangent to a circle only touches one point on the circle and a line from any point on the circle to the center of the circle is a radius. Therefore, any line that is tangent to the circle would also be perpendicular to a radius.(29 votes)

- What is tangent? At0:15.(8 votes)
- A tangent is a line on the outside of a circle or curve that only touches at one point.(20 votes)

- Do opposite angles in a quadrilateral always supplemental (add up to 180 degrees)?(6 votes)
- In general, the answer is no. However, all cyclic quadrilaterals ( quadrilaterals that can have a circumscribed circle drawn) do have opposite angles that add up to 180 degrees. It is a special property of those quadrilaterals, and it can be used to prove that a quadrilateral is cyclic.(11 votes)

- What is a cyclic quadrilateral?(5 votes)
- A cyclic quadrilateral is a quadrilateral that can be inscribed in a circle. In other words, it is a quadrilateral that has a circumcircle. A quadrilateral is cyclic if and only if the opposite angles are supplementary.(6 votes)

- am I the only one or is he speaking really slowly(5 votes)
- If you feel that way , you can change the listening speed to a higher rate!(4 votes)

- For any given arc on the circle, will there be only one possible circumscribed angle?(5 votes)
- At2:11, Sal said the sides are going to add up to 360 degrees, but meant its angles.(4 votes)
- Correct he did mean that, nice catch. There is also a little note box correcting him too at the bottom right.(5 votes)

- Is the circumscribed angle always double the inscribed angle(4 votes)
- Why is an inscribed angle double the central angle? I don't get what the arc has to do with it either.(1 vote)
- The three points (𝐵, 𝐶, 𝐷) that form the inscribed angle also form two isosceles triangles together with the circle center 𝑂,

namely △𝐵𝑂𝐷 and △𝐶𝑂𝐷.

Thereby, 𝑚∠𝑂𝐵𝐷 + 𝑚∠𝑂𝐶𝐷 = 𝑚∠𝐵𝐷𝐶

Thus the measures of three of the four interior angles of quadrilateral 𝑂𝐵𝐶𝐷

add up to 2⋅𝑚∠𝐵𝐷𝐶

The fourth interior angle is 360° − 𝑚∠𝐵𝑂𝐶

The sum of all four angles add to 360°, which gives us

2⋅𝑚∠𝐵𝐷𝐶 + 360° − 𝑚∠𝐵𝑂𝐶 = 360°

⇒ 𝑚∠𝐵𝐷𝐶 = (1∕2)⋅𝑚∠𝐵𝑂𝐶

In other words, the measure of the inscribed angle is half the measure of the central angle.(6 votes)

- At1:41, why are angles C and B 90 degrees? I know it was explained in the video, but I didn't really understand it.(1 vote)
- AB and AC are tangents, meaning that they intersect the circle at one and only one point. One property of tangents are that if you draw a radius to the tangent point, the tangent will be perpendicular to that radius.(5 votes)

## Video transcript

Angle A is a circumscribed
angle on circle O. So this is angle
A right over here. Then when they say it's
a circumscribed angle, that means that the
two sides of the angle are tangent to the circle. So AC is tangent to
the circle at point C. AB is tangent to
the circle at point B. What is the measure of angle A? Now, I encourage you
to pause the video now and to try this out on your own. And I'll give you a hint. It will leverage the fact that
this is a circumscribed angle as you could imagine. So I'm assuming you've
given a go at it. So the other piece
of information they give us is that angle D,
which is an inscribed angle, is 48 degrees and it intercepts
the same arc-- so this is the arc that it intercepts,
arc CB I guess you could call it-- it intercepts this
arc right over here. It's the inscribed angle. The central angle that
intersects that same arc is going to be twice
the inscribed angle. So this is going
to be 96 degrees. I could put three markers here
just because we've already used the double marker. Notice, they both intercept
arc CB so some people would say the measure of
arc CB is 96 degrees, the central angle is 96
degrees, the inscribed angle is going to be half
of that, 48 degrees. So how does this help us? Well, a key clue is that angle
is a circumscribed angle. So that means AC and AB are
each tangent to the circle. Well, a line that is
tangent to the circle is going to be perpendicular to
the radius of the circle that intersects the circle
at the same point. So this right over here is
going to be a 90-degree angle, and this right over here is
going to be a 90-degree angle. OC is perpendicular to CA. OB, which is a radius,
is perpendicular to BA, which is a tangent line, and
they both intersect right over here at B. Now, this
might jump out at you. We have a quadrilateral
going on here. ABOC is a quadrilateral,
so its sides are going to add
up to 360 degrees. So we could know, we
could write it this way. We could write the
measure of angle A plus 90 degrees plus another
90 degrees plus 96 degrees is going to be equal
to 360 degrees. Or another way of thinking
about it, if we subtract 180 from both sides, if we
subtract that from both sides, we get the measure of
angle A plus 96 degrees is going to be equal
to 180 degrees. Or another way of
thinking about it is the measure of angle A
or that angle A and angle O right over here-- you
could call it angle COB-- that these are going to be
supplementary angles if they add up to 180 degrees. So if we subtract 96
degrees from both sides, we get the measure
of angle A is equal to-- I don't want to make that
look like a less than symbol, let make it-- measure of
angle-- this one actually looks more like a--
measure of angle A is equal to 180 minus 96. Let's see, 180 minus
90 would be 90, and then we subtract another
6 gets us to 84 degrees.