Video Transcript
In this video, we will learn how to
find the positions of points, straight lines, and circles with respect to other
circles. Our aim will be to work out the
length of a line or measure of an angle using our knowledge of circles and also our
angle properties or circle theorems. We will begin by recalling some of
the key properties of a circle.
If we consider the circle drawn,
there are four key lines drawn on it. Line number one is the radius. This is the distance from the
center of a circle to its circumference. Line number two is a chord. This is a line segment whose
endpoints are on the circumference. Line three is the diameter of the
circle. This is a special type of chord as
it passes through the center of the circle. The length of the diameter will
always be twice the length of the radius. Finally, we have line number four,
the tangent. A tangent is any line that
intersects the circle at one point. The tangent will touch the
circumference of the circle. We will now consider some angle
properties and circle theorems.
Any tangent that we draw is at
right angles to the radius. This means that the tangent and
radius make an angle of 90 degrees. Any two tangents drawn from the
same point are equal in length. In our diagram, the line segment
𝑥𝑦 is equal to the line segment 𝑥𝑧. Any two radii of a circle are also
equal in length. This is because the radius is the
distance from the center to the circumference of the circle. By drawing a chord as shown on this
diagram, we can create a triangle. As two sides of the triangle are
equal in length, we have an isosceles triangle. This means that the measure of the
two angles in pink are equal.
We recall that the angles in any
triangle sum to 180 degrees. This means that if we are given one
angle in an isosceles triangle, we can calculate the other two. We also recall that angles on a
straight line also sum to 180 degrees. Finally, we will need to use in
this video the alternate segment theorem, also sometimes known as the tangent-chord
theorem. This states that, in any circle,
the angle between a chord and a tangent through one of the endpoints of the chord is
equal to the angle in the alternate segment. This can be seen in the diagram
shown where we have a tangent drawn and a triangle inside the circle where each of
its vertices are on the circumference.
The two pink angles shown will be
equal. The same is true of the two blue
angles. The angle between the chord and the
tangent is equal to the angle in the alternate segment. By considering the third angle in
the triangle in green, we can see that the blue angle plus the pink angle plus the
green angle must sum to 180 degrees, as angles in a triangle and angles on a
straight line sum to 180 degrees. We will now look at some questions
that can be answered using the properties discussed.
The radii of the two circles on the
common center 𝑀 are three centimeters and four centimeters. What is the length of the line
segment 𝐴𝐵?
We are told in the question that
the radius of the smaller circle is three centimeters. Therefore, the length 𝐴𝑀 is equal
to three centimeters. The radius is the distance from the
center of a circle to its circumference. So this means that the line
segments 𝐵𝑀 and 𝐶𝑀 are also equal to three centimeters. We are told that the larger circle
has a radius of four centimeters. This means that the line segment
𝐷𝑀 is equal to four centimeters. In this question, we are asked to
calculate the length of the line segment 𝐴𝐵.
It appears that this is the
diameter of the circle as the line 𝐴𝐵 passes through the center 𝑀. As the diameter is twice the length
of the radius, the line segment 𝐴𝐵 is equal to two times the line segment
𝐴𝑀. We need to multiply three
centimeters by two. This is equal to six
centimeters. The line segment 𝐴𝐵 is equal to
six centimeters. Even if 𝐴𝐵 was not the diameter
of the circle, that is, it was not a straight line, we could calculate its length by
adding the distance 𝐴𝑀 to the distance 𝑀𝐵. As both of these are radii of the
smaller circle, we need to add three centimeters and three centimeters. This once again gives us an answer
of six centimeters.
In our next question, we will use
the properties of tangents and isosceles triangles.
Given that the measure of angle
𝑍𝑌𝐿 is equal to 122 degrees, find the measure of angle 𝑋.
In our diagram, we have two
tangents starting from the point 𝑋. The top one touches the circle at
point 𝑍 and the bottom one touches the circle at point 𝑌. We also have a chord drawn on the
circle from point 𝑍 to point 𝑌. We are told that the measure of
angle 𝑍𝑌𝐿 is 122 degrees. We recall that two tangents drawn
from the same point must be equal in length. This means that the line segment
𝑋𝑍 is equal to the line segment 𝑋𝑌. Triangle 𝑋𝑌𝑍 is therefore
isosceles, as it has two equal length sides. In any isosceles triangle, the
measure of two angles are equal. In this case, angle 𝑋𝑌𝑍 is equal
to angle 𝑋𝑍𝑌.
Angles on a straight line sum to
180 degrees. This means that we can calculate
the measure of angle 𝑋𝑌𝑍 by subtracting 122 from 180. This is equal to 58 degrees. Angles 𝑋𝑌𝑍 and 𝑋𝑍𝑌 are both
equal to 58 degrees. Our aim in this question is to find
the measure of angle 𝑋, and we know that angles in a triangle also sum to 180
degrees. Angle 𝑋 is therefore equal to 180
minus 58 plus 58. 58 plus 58 is equal to 116, and
subtracting this from 180 gives us 64. The measure of angle 𝑋 is
therefore equal to 64 degrees.
In our next question, we will use
the alternate segment theorem.
Given that 𝐵𝐶 is a tangent to the
circle with center 𝑀 and the measure of angle 𝐴𝑀𝐷 is 97 degrees, find the
measure of angle 𝐶𝐵𝐷.
We know that in any circle, a
tangent and radius or tangent and diameter meet at 90 degrees. We are told in the question that
the measure of angle 𝐴𝑀𝐷 is 97 degrees. We need to calculate the measure of
the angle 𝐶𝐵𝐷. The alternate segment theorem
states that, in any circle, the angle between a chord and a tangent through one of
the endpoints of the chord is equal to the angle in the alternate segment. In this question, the measure of
angle 𝐶𝐵𝐷 is equal to angle 𝐵𝐴𝐷.
Let’s now consider how we can
calculate the angle 𝐵𝐴𝐷 using the information on the diagram. Triangle 𝑀𝐴𝐷 is isosceles as the
line segment 𝑀𝐴 is equal to the line segment 𝑀𝐷. They are both radii of the
circle. This means that the angles inside
the triangle, angle 𝑀𝐴𝐷 and angle 𝑀𝐷𝐴, are equal. These can be calculated by
subtracting 97 from 180 and then dividing by two. 180 minus 97 is equal to 83. Dividing this by two or halving it
gives us 41.5. Angles 𝑀𝐴𝐷 and 𝑀𝐷𝐴 are both
equal to 41.5 degrees. As the angle 𝑀𝐴𝐷 is the same as
the angle 𝐵𝐴𝐷, we can see, using the alternate segment theorem, that the measure
of angle 𝐶𝐵𝐷 is also 41.5 degrees.
In our next two questions, we will
need to use algebra to calculate the missing values.
Given that 𝐴𝐷 is a tangent to the
circle and the measure of angle 𝐷𝐴𝐶 is 90 degrees, calculate the measure of angle
𝐴𝐶𝐵.
We are told in the question that
𝐴𝐷 is a tangent and the measure of angle 𝐷𝐴𝐶 is 90 degrees. We know that the tangent to any
circle is perpendicular to the radius or diameter. This means that, in this question,
𝐴𝐶 is a diameter of the circle. We could use two possible angle
properties or circle theorems to solve this problem. Firstly, we could use the fact that
the angle in a semicircle equals 90 degrees. This means that the measure of
angle 𝐴𝐵𝐶 is 90 degrees. We could also have found this using
the alternate segment theorem, where the measure of angle 𝐷𝐴𝐶 is equal to the
measure of angle 𝐴𝐵𝐶. Either way, we know that 𝐴𝐵𝐶 is
equal to 90 degrees.
We now need to solve the equation
nine 𝑥 is equal to 90. Dividing both sides of this
equation by nine gives us 𝑥 is equal to 10. Our value of 𝑥 is 10 degrees. We can see on the diagram that the
measure of angle 𝐴𝐶𝐵 is five 𝑥. As 𝑥 is equal to 10 degrees, we
need to multiply this by five. Five multiplied by 10 is equal to
50. Therefore, angle 𝐴𝐶𝐵 equals 50
degrees.
In our final question, we will
introduce a new theorem called the tangent-secant theorem.
In the figure, two circles with
centers 𝑀 and 𝑁 touch externally at 𝐴, which is a point on the common tangent 𝐿,
where the line segment 𝐴𝐵 is a common tangent. Suppose 𝐴𝐵 is equal to 𝑀𝑁 which
is equal to 45.5 centimeters and 𝐵𝐶 is equal to 30.5 centimeters. Find 𝐴𝑁 to the nearest tenth.
We are told in the question that
the lengths of 𝐴𝐵 and 𝑀𝑁 are both equal to 45.5 centimeters. 𝑀𝑁 is the sum of the radii of the
two circles. It is 𝑟 one plus 𝑟 two, where 𝑟
one is the length of 𝐴𝑁 and 𝑟 two is the length 𝐴𝑀. It is the value of 𝐴𝑁 or 𝑟 one
that we’re trying to find in this question. We are also told that the length of
𝐵𝐶 is 30.5 centimeters. If we consider the right-angled
triangle 𝐴𝐵𝐷, we could use the Pythagorean theorem to calculate a missing
length. This states that 𝑎 squared plus 𝑏
squared is equal to 𝑐 squared, where 𝑐 is equal to the length of the longest side
or hypotenuse.
In this question, 𝐵𝐴 squared plus
𝐴𝐷 squared is equal to 𝐵𝐷 squared. We know that the length of 𝐵𝐴 or
𝐴𝐵 is 45.5 centimeters. However, we currently don’t know
the length of the other two sides. We will, however, be able to
calculate the length of 𝐵𝐷 by using the tangent-secant theorem. This states that the product of the
lengths of the secant segment and its external segment equals the square of the
length of the tangent segment. The external segment is length
𝐵𝐶, the secant segment is 𝐵𝐷, and the tangent segment is 𝐵𝐴. Therefore, 𝐵𝐶 multiplied by 𝐵𝐷
is equal to 𝐵𝐴 squared. Substituting in the values we know,
30.5 multiplied by 𝐵𝐷 is equal to 45.5 squared. Dividing both sides of this
equation by 30.5 gives us a value of 𝐵𝐷 equal to 67.877 and so on. We can now substitute this value
back into the Pythagorean theorem. 45.5 squared plus 𝐴𝐷 squared is
equal to 67.877 squared.
We will now clear some room so we
can continue this calculation. Subtracting 45.5 squared from both
sides gives us 𝐴𝐷 squared is equal to 67.877 squared minus 45.5 squared. Typing the right-hand side into our
calculator gives us 2537.043 and so on. We can then square root both sides
of this equation so that 𝐴𝐷 is equal to 50.369 and so on. 𝐴𝐷 is the diameter of the larger
circle. We can therefore calculate the
radius 𝐴𝑀 by dividing this value by two. This is equal to 25.184 and so
on. Rounding this to one decimal place
or the nearest tenth gives us 25.2 centimeters. The radius 𝐴𝑀 is 25.2
centimeters.
We can now use the fact that 𝑀𝑁
is equal to 45.5 centimeters to calculate the length of 𝐴𝑁. 𝐴𝑁 will be equal to 𝑀𝑁 minus
𝐴𝑀. We need to subtract 25.2 from
45.5. This is equal to 20.3
centimeters. The length of 𝐴𝑁 to the nearest
tenth is 20.3 centimeters.
We will now summarize the key
points from this video. We can use the radius and diameter
as well as any chords and tangents to solve problems involving unknown lengths and
angles in a circle. As we saw in our last question, the
Pythagorean theorem and the tangent-secant theorem can be used to calculate missing
lengths. Angle properties and circle
theorems can be used to calculate missing angles. These include the facts that angles
in a triangle sum to 180 degrees, two angles in an isosceles triangle are equal, a
tangent meets the radius at 90 degrees, the angle in a semicircle is 90 degrees, and
the alternate segment theorem, which states the angle between a chord and a tangent
through one of the endpoints of the chord is equal to the angle in the alternate
segment. There are also several other circle
theorems that were not covered in this video.
