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 Angle in a square (Posted on 2010-11-16)
Points P and Q are respectively located on the sides AB and BC of square ABCD, such that BP = BQ.

H is the base of the perpendicular from point B to the segment PC.

Determine the measure of DHQ.

 No Solution Yet Submitted by K Sengupta Rating: 3.0000 (1 votes)

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 Solution | Comment 1 of 5
`The angle DHQ is 90 degrees.`
`PROOF:`
`Let ABCD be a unit square,`
`   |BP| = |BQ| = b,`
`and F the base of the perpendicularfrom H to side AB.`
`We can then assign co-ordinate pairsto the eight point:`
`   A(1,0)         P(b,0)   B(0,0)         Q(0,b)   C(0,1)         H(f,h)   D(1,1)         F(f,0)`
`From the similar right trianglesPBC and HFB,`
`    1     |CB|     |BF|     f   --- = ------ = ------ = ---    b     |PB|     |HF|     h`
`                or`
`              h = b*f            `
`From the similar right trianglesPBC and PFH,`
`    1     |CB|     |HF|      h      b*f   --- = ------ = ------ = ----- = -----    b     |PB|     |PF|     b-f     b-f`
`                or`
`        b-b^2*f = f         `
`                        b-h     1-h  slope(HQ)*slope(HD) = ----- * -----                          0-f     1-f                                                  b-b*f     1-b*f                     = ------- * -------                          -f       1-f`
`                        b-b^2*f                     = ---------                           -f`
`                     = -1`
`Therefore, HQ and HD are perpendicular.`
` `

 Posted by Bractals on 2010-11-16 21:02:56

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