Class 12 Exam > Class 12 Videos > Example with tangent and radius - Math, Class 12
Example with tangent and radius - Math, Class 12 Video Lecture
FAQs on Example with tangent and radius - Math, Class 12 Video Lecture
| 1. What is the definition of tangent in mathematics? |  |
Ans. In mathematics, a tangent is a straight line that touches a curve at a single point without crossing it at that point. It is perpendicular to the radius of the curve at the point of contact.
| 2. How is the radius related to a tangent in geometry? | |
Ans. The radius of a circle or a sphere is perpendicular to the tangent drawn at the point of contact. This means that the radius and the tangent are always at a right angle to each other.
| 3. Can a tangent be drawn from any point outside a circle? | |
Ans. No, a tangent can only be drawn from a point outside a circle if the point lies on the circumference of the circle. If the point is inside the circle, no tangent can be drawn.
| 4. What is the significance of tangents in real-life applications? | |
Ans. Tangents have various applications in real-life scenarios. For example, in physics, tangents are used to calculate the rate of change of a quantity. In engineering, tangents are used to determine the slope of a curve. In navigation, tangents are used to calculate the direction of a ship or aircraft.
| 5. How can tangents be used to find the length of a radius? | |
Ans. By drawing a tangent to a circle from an external point, we can form a right-angled triangle. Using the Pythagorean theorem, we can relate the lengths of the tangent, the radius, and the distance between the external point and the center of the circle. By rearranging the equation, we can solve for the length of the radius.
Text Transcript from Video
Line AC is tangent to
circle O at point C.
So this is line AC, tangent
to circle O at point
C. What is the
length of segment AC?
What is this distance right over
here, between point A and point
C?
And I encourage you
now to pause this video
and try this out on your own.
So I'm assuming you've
given a go at it.
So the key thing
to realize here,
since AC is tangent to the
circle at point C, that
means it's going to be
perpendicular to the radius
between the center of
the circle and point C.
So this right over
here is a right angle.
And the reason
why that is useful
is now we know that triangle
AOC is a right triangle.
So if we know two
of its sides, we
could use the
Pythagorean theorem
to figure out the third.
Now, we clearly know OC.
Now OA, we don't
know the entire side.
They only give us
that AB is equal to 2.
But the thing that might
jump out in your mind
is OB is a radius.
It's going to be the same
length as any radius.
So this is going
to be 3 as well.
It's the distance between
the center of the circle
and a point on the circle, just
like the distance between O
and C. So this is
going to be 3 as well.
And so now we are
able to figure out
that the hypotenuse of
this triangle has length 5.
And so we need to figure out
what the length of segment AC
is.
So let's just call
that, I don't know.
I'll call that x.
And so we know that x
squared plus 3 squared--
I'm just applying the
Pythagorean theorem here--
is going to be equal to the
length of the hypotenuse
squared, is going to
be equal to 5 squared.
And I know this
is the hypotenuse.
It's the side opposite
the 90-degree angle.
It's the longest side
of the right triangle.
So x squared plus
9 is equal to 25.
Subtract 9 from
both sides, and you
get x squared is equal to 16.
And so it should jump
out at you that x
is going to be equal to 4.
So x is equal to 4.
x is the same thing as
the length of segment AC,
so the length of
segment AC is 4.
circle O at point C.
So this is line AC, tangent
to circle O at point
C. What is the
length of segment AC?
What is this distance right over
here, between point A and point
C?
And I encourage you
now to pause this video
and try this out on your own.
So I'm assuming you've
given a go at it.
So the key thing
to realize here,
since AC is tangent to the
circle at point C, that
means it's going to be
perpendicular to the radius
between the center of
the circle and point C.
So this right over
here is a right angle.
And the reason
why that is useful
is now we know that triangle
AOC is a right triangle.
So if we know two
of its sides, we
could use the
Pythagorean theorem
to figure out the third.
Now, we clearly know OC.
Now OA, we don't
know the entire side.
They only give us
that AB is equal to 2.
But the thing that might
jump out in your mind
is OB is a radius.
It's going to be the same
length as any radius.
So this is going
to be 3 as well.
It's the distance between
the center of the circle
and a point on the circle, just
like the distance between O
and C. So this is
going to be 3 as well.
And so now we are
able to figure out
that the hypotenuse of
this triangle has length 5.
And so we need to figure out
what the length of segment AC
is.
So let's just call
that, I don't know.
I'll call that x.
And so we know that x
squared plus 3 squared--
I'm just applying the
Pythagorean theorem here--
is going to be equal to the
length of the hypotenuse
squared, is going to
be equal to 5 squared.
And I know this
is the hypotenuse.
It's the side opposite
the 90-degree angle.
It's the longest side
of the right triangle.
So x squared plus
9 is equal to 25.
Subtract 9 from
both sides, and you
get x squared is equal to 16.
And so it should jump
out at you that x
is going to be equal to 4.
So x is equal to 4.
x is the same thing as
the length of segment AC,
so the length of
segment AC is 4.
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Dec 02, 2024 Last updated |
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