SOLUTION: Verify that A = (7, 4), B = (−7, 4), C = (−1, −8), and D = (8, −1) all lie on a circle centered at the origin. Let K be the intersection of chords AC and BD. Prove that tri

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Question 1158190: Verify that A = (7, 4), B = (−7, 4), C = (−1, −8), and D = (8, −1) all lie on a circle centered at the origin. Let K be the intersection of chords AC and BD. Prove that triangles KAB and KDC are similar and find the ratio of similarity. Then, show that KA · KC = KB · KD.
Answer by greenestamps(13198) About Me  (Show Source):
You can put this solution on YOUR website!


(1) For each of the points, the distance from the origin is sqrt(65).

(2) Angles A and D are congruent because they both cut off arc BC; angles B and C are congruent because they both cut off arc AD. Two pairs of congruent angles makes the triangles similar.

(3) The lengths of corresponding sides AB and CD are 14 and sqrt(130), so the ratio of similarity is 14:sqrt(130).

(4) KA*KC = KB*KD by the theorem about the lengths of the pieces of two intersecting chords. Or you can prove it using the two similar triangles.