In a circle, chords AB and CD intersect at an interior point P. If AP = 4, PB = 6, and CP = 3, find

Question

In a circle, chords AB and CD intersect at an interior point P. If AP = 4, PB = 6, and CP = 3, find the length of PD.

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{"content":[{"type":"paragraph","content":[{"text":"Step 1:","type":"text","marks":[{"type":"bold"}]},{"text":" By the Intersecting Chords Theorem, when two chords of a circle intersect at an interior point, the products of the segments of the chords are equal:","type":"text"}]},{"attrs":{"latex":"AP \cdot PB = CP \cdot PD"},"type":"math_block"},{"type":"paragraph","content":[{"marks":[{"type":"bold"}],"text":"Step 2:","type":"text"},{"text":" We are given ","type":"text"},{"attrs":{"latex":"AP = 4"},"type":"math_inline"},{"text":", ","type":"text"},{"attrs":{"latex":"PB = 6"},"type":"math_inline"},{"type":"text","text":", and "},{"type":"math_inline","attrs":{"latex":"CP = 3"}},{"text":". Substitute these values into the theorem:","type":"text"}]},{"type":"math_block","attrs":{"latex":"4 \cdot 6 = 3 \cdot PD \implies 24 = 3 \cdot PD"}},{"content":[{"marks":[{"type":"bold"}],"text":"Step 3:","type":"text"},{"text":" Solve for ","type":"text"},{"attrs":{"latex":"PD"},"type":"math_inline"},{"text":":","type":"text"}],"type":"paragraph"},{"attrs":{"latex":"PD = \frac{24}{3} = 8"},"type":"math_block"},{"content":[{"text":"So the length of ","type":"text"},{"attrs":{"latex":"PD"},"type":"math_inline"},{"text":" is ","type":"text"},{"attrs":{"latex":"8"},"type":"math_inline"},{"type":"text","text":"."}],"type":"paragraph"}],"type":"doc"}

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