Welcome to a lesson on solving for unknown values
using the properties of tangents to a circle from a single point.
Remember tangent segments to a circle from a point are congruent.
So in this diagram here.
These two red segments are congruent.
Let's go and take a look at our examples.
Here we want to determine the perimeter of this triangle.
So if we start at point A.
segment AE would be congruent to segment AD.
So this would be six units.
Looking at point B segment BE is congruent to segment BF.
So the length of BF must be four units.
And then considering point C.
segment CF is congruent to segment CD.
Therefore segment CD must have a length of seven units.
So now we just need to add the lengths of all these segments.
Here we have 12
eight
and 14.
So 12 plus eight plus 14.
It's going to give us four, carry the one, that will give us 34 units
as the perimeter of this triangle.
Here's our second example.
And again if we consider point B.
Segment
AB
is going to be congruent to segment CB.
Because they're both tangent to the circle.
And they share the end point B.
So we can set up an equation to solve for x. 4x minus nine
must equal 15
since the segments are congruent.
So now we add nine to both sides.
So we have four x, this is zero.
15 plus nine equals 24.
Divide both sides by four.
We have x equals six.
And we can go ahead and check this.
Four times x or four times six is 24 minus nine.
Does give us 15.
Let's take a look at one more example that's a little bit more challenging.
Notice that segment
EB is tangent to the larger circle.
And segment AB is tangent to the smaller circle.
And notice how they also provided us with the radii
that are perpendicular to the tangent segments.
So we have a right angle here and we have a right angle here.
So we do have two right triangles.
Our goal here is to determine the length of
segment
DC
which is the length of this
entire segment here.
We're not going to be able to do this all in one step.
If we take a look at this largest circle and larger right triangle.
We can determine the length of this segment here
by using Pythagorean Theorem.
So let's go ahead and call this length x.
And then we'll call this length over here y.
Again let's go ahead and solve for x using the Pythagorean Theorem.
We have c squared must equal a squared plus b squared.
Remember c must be the hypotenuse so in this case
x would be c
and the legs a and b would be 10 and 24.
So we're going to have x squared
must equal
a squared which is 10 squared
plus 24 squared.
So now we go ahead and simplify and solve. We have x squared equals
this would be 100
plus
576.
So x squared equals
676.
And now we'll square root both sides of the equation.
676 is a perfect square.
It's equal to 26.
So the length of DB.
Is equal to 26 units. Let's go ahead and write that down.
But we still have to determine the length of
BC which we labeled y.
Again looking back at these triangles.
Notice angle DBE
this angle here.
And angle CBA would be congruent
because they're vertical angles.
So these two right triangles are actually similar triangles.
By angle angle similarity since we know these other two angles are right angles
which measure 90 degrees.
So we can set up a proportion to determine the length of BC
and then add it to the length of DB
to determine the length of DC.
So let's go ahead and do comparison comparing the small
right triangle to the large right triangle.
So the ratio of five to 10.
Must equal
the ratio of y
to
this length here which we found to be 26.
So now we go ahead and cross multiply
and solve for y.
So we have ten y
equals
five times 26, it's going to be 130.
Divide both sides by 10.
We have y equals
13.
Which tells us the length of BC.
Which tells us the length of segment BC
is equal to
13 units.
Now we have all the information we need the length of segment DC
is going to be equal to the sum of
the length of DB plus the length of BC. So 26 plus 13,
it's going to give us 39 units
for the length that we're trying to determine.
Here in orange.
And we're going to stop here for this video.
Thank you for watching.