Video: Positions of Points, Straight Lines, and Circles with Respect to Circles

In this video, we will learn how to find the positions of points, straight lines, and circles with respect to other circles.

17:40

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 π‘₯𝑦 is equal to the line segment π‘₯𝑧. 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 𝐴𝐡?

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 𝐡𝑀 and 𝐢𝑀 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 𝐴𝐡.

It appears that this is the diameter of the circle as the line 𝐴𝐡 passes through the center 𝑀. As the diameter is twice the length of the radius, the line segment 𝐴𝐡 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 𝐴𝐡 is equal to six centimeters. Even if 𝐴𝐡 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 𝑀𝐡. 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 π‘π‘ŒπΏ 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 π‘Œ. We also have a chord drawn on the circle from point 𝑍 to point π‘Œ. We are told that the measure of angle π‘π‘ŒπΏ 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 π‘‹π‘Œ. Triangle π‘‹π‘Œπ‘ 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 π‘‹π‘Œπ‘ is equal to angle π‘‹π‘π‘Œ.

Angles on a straight line sum to 180 degrees. This means that we can calculate the measure of angle π‘‹π‘Œπ‘ by subtracting 122 from 180. This is equal to 58 degrees. Angles π‘‹π‘Œπ‘ and π‘‹π‘π‘Œ 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 𝐡𝐢 is a tangent to the circle with center 𝑀 and the measure of angle 𝐴𝑀𝐷 is 97 degrees, find the measure of angle 𝐢𝐡𝐷.

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 𝐢𝐡𝐷. 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 𝐢𝐡𝐷 is equal to angle 𝐡𝐴𝐷.

Let’s now consider how we can calculate the angle 𝐡𝐴𝐷 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 𝐡𝐴𝐷, we can see, using the alternate segment theorem, that the measure of angle 𝐢𝐡𝐷 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 𝐷𝐴𝐢 is 90 degrees, calculate the measure of angle 𝐴𝐢𝐡.

We are told in the question that 𝐴𝐷 is a tangent and the measure of angle 𝐷𝐴𝐢 is 90 degrees. We know that the tangent to any circle is perpendicular to the radius or diameter. This means that, in this question, 𝐴𝐢 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 𝐴𝐡𝐢 is 90 degrees. We could also have found this using the alternate segment theorem, where the measure of angle 𝐷𝐴𝐢 is equal to the measure of angle 𝐴𝐡𝐢. Either way, we know that 𝐴𝐡𝐢 is equal to 90 degrees.

We now need to solve the equation nine π‘₯ is equal to 90. Dividing both sides of this equation by nine gives us π‘₯ is equal to 10. Our value of π‘₯ is 10 degrees. We can see on the diagram that the measure of angle 𝐴𝐢𝐡 is five π‘₯. As π‘₯ is equal to 10 degrees, we need to multiply this by five. Five multiplied by 10 is equal to 50. Therefore, angle 𝐴𝐢𝐡 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 𝐴𝐡 is a common tangent. Suppose 𝐴𝐡 is equal to 𝑀𝑁 which is equal to 45.5 centimeters and 𝐡𝐢 is equal to 30.5 centimeters. Find 𝐴𝑁 to the nearest tenth.

We are told in the question that the lengths of 𝐴𝐡 and 𝑀𝑁 are both equal to 45.5 centimeters. 𝑀𝑁 is the sum of the radii of the two circles. It is π‘Ÿ one plus π‘Ÿ two, where π‘Ÿ one is the length of 𝐴𝑁 and π‘Ÿ two is the length 𝐴𝑀. It is the value of 𝐴𝑁 or π‘Ÿ one that we’re trying to find in this question. We are also told that the length of 𝐡𝐢 is 30.5 centimeters. If we consider the right-angled triangle 𝐴𝐡𝐷, we could use the Pythagorean theorem to calculate a missing length. This states that π‘Ž 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, 𝐡𝐴 squared plus 𝐴𝐷 squared is equal to 𝐡𝐷 squared. We know that the length of 𝐡𝐴 or 𝐴𝐡 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 𝐡𝐷 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 𝐡𝐢, the secant segment is 𝐡𝐷, and the tangent segment is 𝐡𝐴. Therefore, 𝐡𝐢 multiplied by 𝐡𝐷 is equal to 𝐡𝐴 squared. Substituting in the values we know, 30.5 multiplied by 𝐡𝐷 is equal to 45.5 squared. Dividing both sides of this equation by 30.5 gives us a value of 𝐡𝐷 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.

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