Video Transcript
In this video, we will learn how to
use the properties of tangents of circles to find the missing angles or side
lengths. This is part of a wider subject
with circle theorems, which focuses on properties of angles formed inside circles by
chords, tangents, and radii. In this lesson, we’ll be
specifically focusing on properties of the angles and lengths made by tangents drawn
from exterior points to the circumference of a circle. You should, however, be familiar
with general angle properties such as the sum of angles on a straight line and the
sum of angles in a triangle.
Remember, first of all, that a
tangent to the circle is a straight line that intersects the circle at a single
point. It doesn’t pass inside the circle,
but instead just meets the circle at a point on its circumference.
The first key property we’re going
to consider is this: a tangent to a circle is perpendicular to the radius at the
point of contact. What this means is that if we draw
in the radius of the circle from the point on the circumference where the tangent
touches the circle, then the angle between the tangent and the radius will be a
right angle. Now, of course, it’s also true that
the tangent will be perpendicular to the diameter of the circle at this point, as
this is just a continuation of the radius. But it’s the radius that we tend to
use when quoting this result. Now, the proof of this requires
some of the other circle theorems, including one called the alternate segment
theorem. So, we won’t go into it here. However, we will see a proof of
another key property later in this video. So, we’ll still get a flavor of how
to prove these theorems.
Let’s now consider some examples
using this first property.
Given that the line segment 𝐴𝐵 is
a tangent to the circle 𝑀, and the measure of angle 𝑀𝐵𝐹 is 123 degrees,
determine the measure of angle 𝐴𝑀𝐵.
Angle 𝐴𝑀𝐵 is the angle formed
when we travel from 𝐴 to 𝑀 to 𝐵. So, it’s the angle marked in orange
on the figure. Angle 𝑀𝐵𝐹 is the angle made when
we travel from 𝑀 to 𝐵 to 𝐹. So, it’s the angle now labelled in
pink, and its measure is 123 degrees. We can see that the angle we’re
looking to find — angle 𝐴𝑀𝐵 — is contained within a triangle. If we can work out the other two
angles in this triangle, then we can use the fact that the angle sum in any triangle
is 180 degrees to find the angle we’re looking for.
Firstly, let’s consider the angle
𝑀𝐵𝐴. One of our most basic angle facts
is that angles on a straight line sum to 180 degrees. And this angle is on a straight
line with the angle we’ve already marked as being 123 degrees. So, we can say that angle 𝑀𝐵𝐹
plus angle 𝑀𝐵𝐴 equals 180 degrees. As stated, we already know the
measure of angle 𝑀𝐵𝐹. So, we can substitute this
value. And we now have an equation we can
solve to find the measure of angle 𝑀𝐵𝐴. We need to subtract 123 from each
side of this equation. Doing so and we find that angle
𝑀𝐵𝐴 is 57 degrees.
So, we found one of the angles in
triangle 𝑀𝐵𝐴. Can we find another one? What about angle 𝑀𝐴𝐵? Well, this is the angle formed
where a tangent to the circle — that’s the line 𝐴𝐵 — meets the radius of the
circle 𝐴𝑀. And we know that a tangent to a
circle is perpendicular to the radius at the point of contact. So, we know that angle 𝑀𝐴𝐵 is 90
degrees. It’s a right angle. We’ve, therefore, found two of the
angles in triangle 𝐴𝐵𝑀. And using the angle sum in a
triangle, we can find the third.
We have the equation angle 𝐴𝑀𝐵
plus 90 degrees plus 57 degrees equals 180 degrees. 90 plus 57 is 147. And subtracting 147 degrees from
each side of the equation gives angle 𝐴𝑀𝐵 equals 33 degrees. So, by using two basic angles facts
as well as the key result that a tangent to a circle is perpendicular to the radius
at the point of contact, we found that the measure of angle 𝐴𝑀𝐵 is 33
degrees.
Let’s now consider a second
application of this important result.
Given that the line segment 𝐴𝐵 is
tangent to the circle 𝑀 at 𝐴, 𝐴𝑀 equals 8.6 centimeters, and 𝑀𝐵 equals 12.3
centimeters, find the length of the line segment 𝐴𝐵, and round the result to the
nearest tenth.
Let’s begin by adding the
information given in the question onto the diagram. 𝐴𝑀 is 8.6 centimeters. That’s this length here. 𝑀𝐵 is 12.3 centimeters. That’s this length here. And the length we’re looking to
find is the length of the line segment 𝐴𝐵. Now, we notice that we have a
triangle, triangle 𝐴𝑀𝐵, in which we know the lengths of two of the sides. Your first thought then may be that
we could apply the Pythagorean theorem. But remember, the Pythagorean
theorem is only valid in right triangles. So, we need to consider whether the
triangle 𝐴𝑀𝐵 is a right triangle.
The other key piece of information
given in the question is that the line segment 𝐴𝐵 is tangent to the circle 𝑀 at
𝐴. A key property about tangents to
circles is that a tangent to a circle is perpendicular to the radius at the point of
contact. So, the line segment 𝐴𝐵 is
perpendicular to the radius 𝐴𝑀. And we, therefore, have a right
angle at 𝐴 in our triangle 𝐴𝑀𝐵. We do indeed have a right
triangle. And so, we can apply the
Pythagorean theorem to calculate the length of the third side.
The Pythagorean theorem tells us
that in a right triangle, the sum of the squares of the two shorter sides which we
can think of as 𝑎 and 𝑏 is equal to the square of the longest side of the
triangle, which we can think of as 𝑐. Remember, the longest side or
hypotenuse is always the side directly opposite the right angle. So, in this case, that’s the side
𝑀𝐵. Substituting 𝐴𝐵 and 8.6 for the
two shorter sides of the triangle and 12.3 for the longest side or hypotenuse, we
have the equation 𝐴𝐵 squared plus 8.6 squared equals 12.3 squared. We can evaluate 8.6 squared and
12.3 squared and then subtract 73.96 — that’s 8.6 squared — from each side, giving
𝐴𝐵 squared equals 77.33.
We solve this equation by square
rooting. And we’re only going to take the
positive value here as 𝐴𝐵 has a physical meaning as the length of a side in this
triangle. Evaluating this square root on a
calculator, we find that 𝐴𝐵 is equal to 8.79374. Remember, though, that we were
asked to around the result to the nearest tenth. So, as there is a nine in the
hundredths column, we round up, giving 𝐴𝐵 equals 8.8 centimeters.
So in this problem, by applying the
key property that a tangent to a circle is perpendicular to the radius at the point
of contact, we were able to deduce that triangle 𝑀𝐴𝐵 was a right triangle. And hence, we can apply the
Pythagorean theorem to calculate the length of its third side. We found that the length of 𝐴𝐵 to
the nearest tenth is 8.8 centimeters.
Let’s now look at one final example
of how we can apply this first property.
Given that the line segment 𝐴𝐵 is
a tangent to the circle 𝑀, and the measure of angle 𝐴𝐵𝑀 is 49 degrees, determine
the measure of angle 𝐴𝐷𝐵.
Angle 𝐴𝐷𝐵 is the angle formed
when we travel from 𝐴 to 𝐷 to 𝐵. So, it’s the angle marked in orange
on the diagram. Angle 𝐴𝐵𝑀 is the angle formed
when we travel from 𝐴 to 𝐵 to 𝑀. It’s the angle now marked in pink
on the diagram with its measure of 49 degrees. From the information given, we
aren’t able to calculate angle 𝐴𝐷𝐵 directly. We’re going to need to find the
measures of some other angles in the figure first. The other key piece of information
given in the question, though, is that the line 𝐴𝐵 is a tangent to the circle
𝑀. And the key property about tangents
of circles is that a tangent to a circle is perpendicular to the radius at the point
of contact.
The point where the tangent meets
the circle is point 𝐴. And the radius here is the line
segment 𝐴𝑀. So, we know that the angle 𝐵𝐴𝑀
is 90 degrees. So, we now know one more angle
within the figure. We still aren’t able to calculate
angle 𝐴𝐷𝐵 directly. So, let’s see what other angles we
could work out. We have a triangle. In fact, it’s a right triangle,
triangle 𝐴𝑀𝐵. And we know two of its angles, the
right angle and the angle of 49 degrees. So, using the fact that angles in a
triangle sum to 180 degrees, we can calculate the third angle in this triangle.
We have that angle 𝐴𝑀𝐵 plus 90
degrees plus 49 degrees equals 180 degrees. 90 plus 49 is 139. And subtracting this from 180, we
find that angle 𝐴𝑀𝐵 is 41 degrees. So, we now know another angle in
our diagram. We still don’t have enough
information to calculate angle 𝐴𝐷𝐵. But we can now calculate a
different angle, angle 𝐴𝑀𝐷. We know that the angles on any
straight line sum to 180 degrees. So, angle 𝐴𝑀𝐷 plus the angle
we’ve just calculated of 41 degrees must equal 180 degrees. Angle 𝐴𝑀𝐷 is, therefore, equal
to 180 degrees minus 41 degrees. That’s 139 degrees.
Now, we found almost all of the
angles in the figure, but still not the one that we were looking for. The final step is to consider
triangle 𝐴𝑀𝐷, in which we know one angle of 139 degrees. We need to notice that both 𝑀𝐷
and 𝑀𝐴 are radii of the circle 𝑀. And therefore, they’re of the same
length. This means that triangle 𝑀𝐷𝐴 is
an isosceles triangle. And it also means that angle 𝑀𝐷𝐴
will be equal to angle 𝑀𝐴𝐷. We can, therefore, find the measure
of each angle by subtracting the third angle, 139 degrees, from the total angle sum
in a triangle, 180 degrees, and then halving the remainder. Doing so gives each of these angles
to be 20.5 degrees. Now, angle 𝑀𝐷𝐴 is in fact the
same angle as angle 𝐴𝐷𝐵. They both refer to this angle
here. And so, we’ve completed the
problem.
By using some of the more basic
facts about angles in triangles and angles in straight lines and then the key
property that a tangent to a circle is perpendicular to the radius at the point of
contact, we’ve found the measure of angle 𝐴𝐷𝐵 is 20.5 degrees.
So, we’ve now seen three
applications of the first key property about tangents of circles. Let’s now introduce the second
property. It’s this: tangents drawn to a
circle from the same exterior point are equal in length.
In our diagram, the two tangents
have been drawn from the exterior point 𝐴. So, the length of the line 𝐴𝐵
will be equal to 𝐴𝐶. Now, we can prove this property
using congruent triangles and the first property. We’ll add a couple of lines to our
diagram, firstly, the radii 𝑀𝐵 and 𝑀𝐶 and, secondly, a line connecting our
exterior point 𝐴 to the center of the circle 𝑀. Now, we’re going to consider the
two triangles 𝐴𝐵𝑀 and 𝐴𝐶𝑀.
We know from the first property
that both tangents will be perpendicular to the radius at the point of contact. So, we know that angle 𝐴𝐵𝑀 and
angle 𝐴𝐶𝑀 are each 90 degrees. But in particular, they’re equal to
one another. We also know that the line segments
𝐵𝑀 and 𝐶𝑀 are each radii of the circle. And so, they will be of equal
length. And the line 𝐴𝑀 is a shared side
in these two triangles. In fact, it is the hypotenuse of
the two triangles. We’ve shown then that these two
triangles each have a right angle, they share a common hypotenuse, and they have one
of the two shorter sides of the triangle also equal in length. Therefore, the two triangles are
congruent by the RHS. That’s right-angle-hypotenuse-side
congruency condition. If the two triangles are congruent,
then the lengths of their third sides — that’s 𝐴𝐵 and 𝐴𝐶 — must be the same. And so, we’ve shown that 𝐴𝐵 is
indeed equal to 𝐴𝐶 and thus proven this property.
Let’s look then at one application
of this.
Find the value of 𝑥.
In the diagram, we can see that we
have a circle and then two lines 𝐴𝐵 and 𝐴𝐶, each of which are tangents to the
circle. The two tangents have been drawn
from the same exterior point, point 𝐴. One of the key properties of
tangents of circles is that tangents drawn from the same exterior point are equal in
length. So, we know that the line segment
𝐴𝐵 is equal in length to the line segment 𝐴𝐶. We’ve been given the length of
𝐴𝐵. It’s 21 centimeters. And we’ve been given an expression
for the length of 𝐴𝐶. It’s two 𝑥 plus five
centimeters. So, equating these, we can form an
equation, two 𝑥 plus five is equal to 21.
We can now solve this equation to
determine the value of 𝑥. We first subtract five from each
side, giving two 𝑥 equals 16, and then divide by two, giving 𝑥 equals eight. Using the key property then, the
tangents drawn from the same exterior point are equal in length, we’ve found the
value of 𝑥. 𝑥 is equal to eight.
This example involves solving a
simple linear equation. But more complicated examples of
this type may involve setting up and solving simultaneous equations. The algebra maybe more complicated,
but the principles involved will be the same.
Let’s now review what we’ve seen in
this lesson. We’ve introduced two key properties
of tangents to circles. Firstly, a tangent to a circle is
perpendicular to the radius at the point of contact. And by extension, it’s also
perpendicular to the diameter of the circle at this point. Secondly, we saw and proved that
tangents drawn to a circle from the same exterior point are equal in length. In this diagram, this means that
𝐴𝐵 is equal to 𝐴𝐶. We’ve also seen that we can use
these properties in partnership with other angle rules and the Pythagorean theorem
to find missing angles and side lengths in problems involving tangents to
circles.
