If 𝐸𝐴 over 𝐸𝐵 is equal to eight over seven, 𝐸𝐶 equals seven centimeters, and 𝐸𝐷 equals eight centimeters, find the lengths of line segment 𝐸𝐵 and line segment 𝐸𝐴.
First, let’s fill in what we know. 𝐸𝐶 equals seven centimeters. 𝐸𝐷 equals eight centimeters. We have to think carefully about how we would note 𝐸𝐴 over 𝐸𝐵. This is a ratio relationship. One way we could note this is by saying 𝐸𝐵 is equal to seven 𝑥 and 𝐸𝐴 is equal to eight 𝑥. For example, if 𝐸𝐴 was equal to 16, 𝐸𝐵 would have to be equal to 14 because 14 to 16 is in the ratio of seven to eight.
So, for now, we’ll leave it as seven 𝑥 to eight 𝑥. There’s something else we need to remember about chords to solve this problem. If two chords intersect, we end up with four segments. And the product of the two segments in each chord must be equal to the product of the two segments in the other chord. Here, we have 𝑎 times 𝑏 must be equal to 𝑐 times 𝑑.
And in our problem, that means seven centimeters times eight centimeters must be equal to seven 𝑥 times eight 𝑥. Seven times eight is 56. So, we have 56 centimeters squared on the left. Seven times eight is 56 and 𝑥 times 𝑥 is 𝑥 squared. From there, we divide both sides by 56. 56 divided by 56 is one. If one centimeter squared is equal to 𝑥 squared and we take the square root of both sides, the square root of one centimeter squared would be one centimeter and the square root of 𝑥 would be 𝑥.
Since 𝑥 equals one centimeter, we multiply seven times one centimeter to get seven centimeters. And eight times one centimeter equals eight centimeters. So, 𝐸𝐵 equals seven centimeters and 𝐸𝐴 equals eight centimeters.