ext{ extdollar}OCD ext{ extdollar} is a triangle. ext{ extdollar}A ext{ extdollar} is a point on ext{ extdollar}OC ext{ extdollar} and ext{ extdollar}B ext{ extdollar} is a point on ext{ extdollar}OD ext{ extdollar}. ext{ extdollar}OA = a ext{ extdollar} and ext{ extdollar}OB = b ext{ extdollar}. ext{ extdollar}OA = OC ext{ extdollar} and ext{ extdollar}OB: BD = 3 : 2 ext{ extdollar}. ext{ extdollar}X ext{ extdollar} is a point on ext{ extdollar}OC ext{ extdollar} such that ext{ extdollar}BX ext{ extdollar} is parallel to ext{ extdollar}DC ext{ extdollar}. Find ext{ extdollar}XD ext{ extdollar}. Give your answer as simply as possible in terms of ext{ extdollar}a ext{ extdollar} and ext{ extdollar}b ext{ extdollar}
Q. ext{ extdollar}OCD ext{ extdollar} is a triangle. ext{ extdollar}A ext{ extdollar} is a point on ext{ extdollar}OC ext{ extdollar} and ext{ extdollar}B ext{ extdollar} is a point on ext{ extdollar}OD ext{ extdollar}. ext{ extdollar}OA = a ext{ extdollar} and ext{ extdollar}OB = b ext{ extdollar}. ext{ extdollar}OA = OC ext{ extdollar} and ext{ extdollar}OB: BD = 3 : 2 ext{ extdollar}. ext{ extdollar}X ext{ extdollar} is a point on ext{ extdollar}OC ext{ extdollar} such that ext{ extdollar}BX ext{ extdollar} is parallel to ext{ extdollar}DC ext{ extdollar}. Find ext{ extdollar}XD ext{ extdollar}. Give your answer as simply as possible in terms of ext{ extdollar}a ext{ extdollar} and ext{ extdollar}b ext{ extdollar}
Triangle Similarity: Since OA=OC and triangles OAD and OCB are similar (by AA similarity), we have OCOD=OAOB.
Ratio Calculation: Given OB:BD=3:2, we can express BD as 52 of OB, so BD=52b.
Expression for OD: Since OD=OB+BD, we can write OD=b+52b=57b.
Proportion Calculation: Using the similarity of triangles, we have OD:OC=OB:OA, so (57b):a=b:a.
Error Identification: From the proportion, we get 57b=b, which is not possible since b cannot be equal to 57b unless b is zero, which is not the case here. We made a mistake.
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