Given that 𝐸𝐴 equals 5.2 centimeters, 𝐸𝐶 equals six centimeters, 𝐸𝐵 equals 7.5 centimeters, and 𝐸𝐷 equals 6.5 centimeters, do the points 𝐴, 𝐵, 𝐶, and 𝐷 lie on a circle?
Let’s consider the diagram we’ve been given. It features two line segments, 𝐴𝐵 and 𝐶𝐷, which intersect at a point 𝐸. We’re given in the question the length of the line segment from point 𝐸 to each of the four points 𝐴, 𝐵, 𝐶, and 𝐷. 𝐸𝐴 is 5.2 centimeters, 𝐸𝐶 is six centimeters, 𝐸𝐵 is 7.5 centimeters, and 𝐸𝐷 is 6.5 centimeters. We’re asked to determine whether the points 𝐴, 𝐵, 𝐶, and 𝐷 lie on a circle.
Now this sort of setup as we have in the diagram, two intersecting line segments, should remind us of the intersecting chords theorem. This tells us that if two chords 𝐴𝐵 and 𝐶𝐷 intersect at a point 𝐸, then 𝐴𝐸 multiplied by 𝐸𝐵 is equal to 𝐶𝐸 multiplied by 𝐸𝐷. Essentially, what this is telling us is that the product of the lengths of the two segments that point 𝐸 divides each chord into is the same for both chords. Now the converse of this is also true, which means that if this relationship holds for two intersecting line segments, then those line segments are chords of a circle. So the endpoints of those line segments lie on its circumference.
To answer this question then, we need to test whether this relationship is satisfied by the lengths that we’ve been given. First, for the line segment 𝐴𝐵, 𝐴𝐸 multiplied by 𝐸𝐵, or 𝐸𝐴 multiplied by 𝐸𝐵, is 5.2 multiplied by 7.5, which is equal to 39. Then for the line segment 𝐶𝐷, 𝐶𝐸 multiplied by 𝐸𝐷, or 𝐸𝐶 multiplied by 𝐸𝐷, is six multiplied by 6.5, which is also equal to 39. As the product is the same for both line segments, the intersecting chords theorem is satisfied, and so the two line segments 𝐴𝐵 and 𝐶𝐷 are chords of the same circle.
The points 𝐴, 𝐵, 𝐶, and 𝐷 do lie on a circle, and so our answer to the question is yes.