A (-3, 4), B (3, -1) and C (-2, 4) are the vertices of a triangle ABC. Find the length of line segment AP, where point P lies inside BC, such that BP : PC = 2 : 3.

7 units

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8 units

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5 units

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4 units

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Solution

The correct option is C 5 units
$B P colon P C equals 2 colon 3 C o minus o r d i n a t e s space o f space P space a r e open parentheses fraction numerator 2 cross times left parenthesis negative 2 right parenthesis plus 3 cross times 3 over denominator 2 plus 3 end fraction comma fraction numerator 2 cross times 4 plus 3 cross times left parenthesis negative 1 right parenthesis over denominator 2 plus 3 end fraction close parentheses equals open parentheses fraction numerator negative 4 plus 9 over denominator 5 end fraction comma fraction numerator 8 minus 3 over denominator 5 end fraction close parentheses equals open parentheses 1 comma 1 close parentheses U sin g space d i s tan c e space f o r m u l a space comma w e space h a v e A P equals square root of left parenthesis 1 plus 3 right parenthesis squared plus left parenthesis 1 minus 4 right parenthesis squared end root equals square root of 16 plus 9 end root equals square root of 25 equals 5 space u n i t s$
Hence, the required length of the line segment AP is 5 units.

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