Welcome to a lesson on solving for unknown values using the properties
of a tangent to a circle.
Let's first review that property.
A line is tangent to a circle if and only if the line is perpendicular
to the radius drawn to the point of tangency.
So this is a tangent line
where B is the point of tendency.
AB would be perpendicular to
line BC.
Let's go ahead and take a look at our first example.
Here were given a tangent segment
in a radius of the circle drawn to that point of tangency
which we know formed a right angle.
So now that we know we have a right triangle here.
We can use the Pythagorean theorem to determine the value of x.
So we use the Pythagorean theorem that states that the
sum of the squares of the legs
of a right triangle equal to the square of the hypotenuse of the right triangle.
So this side is the hypotenuse therefore it has to be c
and we let this side
be side a
and this side be b.
So we'll have x squared plus 20 squared
must equal 24 squared.
So now we'll go ahead and simplify this and then solve. So we'll have x squared.
Plus 20 squared is 400.
Equals
24 squared is equal to 576.
24 times 24.
So now we'll subtract 400 on both sides.
That will be zero so we'll have x squared
equals this would be 176.
Now we'll square root both sides of the equation.
So x is the length were only gonna have the principal square root of the positive value.
So we have x equals the square root of 176.
Now we should try to simplify this.
So let's go ahead and see if we can identify any perfect square factors of
176.
This number is divisible by four.
176 is four times 44.
Four is equal to two times two which are both prime.
44 is four times 11.
11 prime.
And then four is again two times two.
So we wanna show some work here we have x equals the square root of
we have four factors of two.
and a factor of 11.
Every time we have two equal factors we have a perfect square factor.
So two times two is four that's a perfect square.
Two times two is four that's a perfect square.
So this pair gives us a two this pair gives us a two.
So we have a four outside the square root.
So four times the square root of 11.
Now let's take a look at our second example.
Again the same type of problem here were given a tangent segment.
And the radius of the circle drawn to the point of tangency.
So we know we have a right angle here.
So now that we have a right triangle we can use the Pythagorean Theorem to determine the value of x.
But notice how now we're going to be solving for the hypotenuse so this will be side c
and we'll call this side a and this side b.
So again the Pythagorean Theorem
we have a squared plus b squared
equals c squared.
So now we're going to have
eight squared plus 15 squared equals x squared.
We'll eight squared
is equal to 64.
15 square is equal to 225.
64 plus 225
equals 289.
And again we're gonna take the square of both sides.
And because it's a length on a triangle.
We're only gonna consider the principal or positive square root
So we have x equals low of the square of 289.
And actually 289 is a perfect square.
It might not be easy to recognize
but 289 is equal to 17 times 17.
So this simplifies nicely x is equal to 17 units.
I hope you found these examples helpful.