Question Video: Understanding the Intersecting Chords Theorem Mathematics • 11th Grade

Given that ๐ธ๐ด = 5.2 cm, ๐ธ๐ถ = 6 cm, ๐ธ๐ต = 7.5 cm, and ๐ธ๐ท = 6.5 cm, do the points ๐ด, ๐ต, ๐ถ, and ๐ท lie on a circle?


Video Transcript

Given that ๐ธ๐ด equals 5.2 centimeters, ๐ธ๐ถ equals six centimeters, ๐ธ๐ต equals 7.5 centimeters, and ๐ธ๐ท equals 6.5 centimeters, do the points ๐ด, ๐ต, ๐ถ, and ๐ท lie on a circle?

Letโ€™s consider the diagram weโ€™ve been given. It features two line segments, ๐ด๐ต and ๐ถ๐ท, which intersect at a point ๐ธ. Weโ€™re given in the question the length of the line segment from point ๐ธ to each of the four points ๐ด, ๐ต, ๐ถ, and ๐ท. ๐ธ๐ด is 5.2 centimeters, ๐ธ๐ถ is six centimeters, ๐ธ๐ต is 7.5 centimeters, and ๐ธ๐ท is 6.5 centimeters. Weโ€™re asked to determine whether the points ๐ด, ๐ต, ๐ถ, and ๐ท 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 ๐ด๐ต and ๐ถ๐ท intersect at a point ๐ธ, then ๐ด๐ธ multiplied by ๐ธ๐ต is equal to ๐ถ๐ธ multiplied by ๐ธ๐ท. Essentially, what this is telling us is that the product of the lengths of the two segments that point ๐ธ 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 weโ€™ve been given. First, for the line segment ๐ด๐ต, ๐ด๐ธ multiplied by ๐ธ๐ต, or ๐ธ๐ด multiplied by ๐ธ๐ต, is 5.2 multiplied by 7.5, which is equal to 39. Then for the line segment ๐ถ๐ท, ๐ถ๐ธ multiplied by ๐ธ๐ท, or ๐ธ๐ถ multiplied by ๐ธ๐ท, 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 ๐ด๐ต and ๐ถ๐ท are chords of the same circle.

The points ๐ด, ๐ต, ๐ถ, and ๐ท do lie on a circle, and so our answer to the question is yes.

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