Video: Finding the Length of a Chord Using the Chords Theorem

Given that the points 𝐴, 𝐡, 𝐢, and 𝐷 lie on a circle, find the length of segment 𝐡𝐴.

01:59

Video Transcript

Given that the points 𝐴, 𝐡, 𝐢, and 𝐷 lie on a circle, find the length of segment 𝐡𝐴.

So we’d like to find this length of the segment. And we are also told that 𝐴, 𝐡, 𝐢, and 𝐷 lie on a circle, so maybe something like this. Since we have two lines containing the two segments 𝐡𝐴 and 𝐢𝐷 intersecting at point 𝐸, we can use this to solve for the length of 𝐸𝐴. And solving for the length of 𝐸𝐴 will help us find the length of 𝐡𝐴.

To do so, we can use the fact that 𝐸𝐴 times 𝐸𝐡 will be equal to 𝐸𝐢 times 𝐸𝐷. And again, finding this length 𝐸𝐴 is what’s going to help us find the length of 𝐡𝐴. So we can go ahead and write 𝐸𝐴 because we don’t know what it is. And the length of 𝐸𝐡 is 36. The length 𝐸𝐢 we can actually find, because all we need to do is take 39 centimetres plus 33 centimetres to give us 72 centimetres. That’s the length of 𝐸𝐢. And then, lastly, we need to multiply by 𝐸𝐷, which is 33 centimetres.

So we can begin solving for 𝐸𝐴 by multiplying 72 and 33 on the right-hand side of the equation. 72 times 33 is 2367. So now to solve for 𝐸𝐴, we divide both sides of the equation by 36. And we find that the length of segment 𝐸𝐴 is 66 centimetres.

And now we can use this to find the length of 𝐡𝐴. And this is because the length of 𝐡𝐴 will be equal to 𝐸𝐴 minus 𝐸𝐡. 𝐸𝐴 we found to be 66 centimetres. 𝐸𝐡 is 36 centimetres. So subtracting those, we find that the length of segment 𝐡𝐴 would be 30 centimetres.

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