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Find the distance between two points whose coordinates are given: Let P 1  and P 2  be the two given points, and let their coordinates be respectively (x 1 , y 1 ) and (x 2 , y 2 ). Draw P 1 M 1  and P 2 M 2  parallel to OY, to meet OX in M 1  and M 2 . Draw P 2 R parallel to OX to meet M 1 P 1  in R. Then, P 2 R = M 2 M 1  = OM 1 -OM 2  = x 1 -x 2 RP 1  = M 1 P 1 - M 2 P 2  = y 1 -y 2 And ∆P 2 RP 1  = ∆OM 1 P 1  = 180⁰ - P 1 M 1 X = 180⁰ - ω We therefore, have P 1 P 2 2 = P 2 R2 + RP 1 2 – 2P 2 R . RP 1  cos P 2 RP 1 = (x 1  – x 2 ) 2  + (y 1  –y 2 ) 2  – 2(x 1 -x 2 )(y 1 -y 2 )cos(180⁰ - ω) = (x 1  – x 2 ) 2  + (y 1  –y 2 ) 2  – 2(x 1 -x 2 )(y 1 -y 2 )cos ω …(1) If the axes be, as is generally the case, at right angles, we have ω = 90⁰ and hence, cos ω = 0. The formula (1), then becomes P 1 P 2 2  = (x 1 -x 2 ) 2  + (y 1  –y 2 ) 2 So th...