Video: Relationships between Circles

The radii of the two circles on the common center 𝑀 are 3 cm and 4 cm. What is the length of line 𝐴𝐡?

02:33

Video Transcript

The radii of the two circles on the common centre 𝑀 are three centimetres and four centimetres. What is the length of line 𝐴𝐡?

Let’s add the information we know to the diagram. The centre of both circles is 𝑀, and this means the distance from 𝑀 to any point on the outside of the smallest circle will be equal to three centimetres. It also means that the distance from 𝑀 to any point on the outside of the largest circle will be equal to four centimetres. And so, the distance from 𝑀 to 𝐷 is four centimetres. And the distance from 𝐢 to 𝐷 must be equal to one centimetre.

But the line we’re interested in is line 𝐴𝐡. That’s this line. We noticed that the point 𝐴 falls on the outside of the smaller circle and point 𝐡 falls on the outside of the smaller circle. We can also see that the line 𝐴𝐡 runs through the centre 𝑀. This means that the segment 𝐴𝑀 measures three centimetres and the segment 𝑀𝐡 measures three centimetres. We could say that line 𝐴𝐡 is equal to line 𝐴𝑀 plus line 𝑀𝐡. Line 𝐴𝑀 measures three centimetres. Line 𝑀𝐡 measures three centimetres. Together, they form the measure of line 𝐴𝐡. And so, we can say that line 𝐴𝐡 is six centimetres long.

Let’s go back to that original sentence. Line 𝐴𝐡 is equal to line 𝐴𝑀 plus line 𝑀𝐡. We know that the first segment is a radius of this circle, and the second segment is a radius of this circle. Line 𝐴𝐡 is made up of two lines that run through the centre of the circle. And that means we can call line 𝐴𝐡 a diameter of the circle. A diameter of the circle is always equal to two times the radius. And so, we can say the length of line 𝐴𝐡 is six centimetres, and the diameter of the smaller circle with centre 𝑀 is six centimetres.

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