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CBSE Class 10 Answered

the three vertices of a 11gm ABCD are A[3,-4] ,B[-1,-3], C[-6,2] . find the coordinates of vertex D and find the are of ABCD.  
Asked by Namanjain6612 | 25 Nov, 2018, 06:55: PM
answered-by-expert Expert Answer
Let coordinates of D be (x,y)
 
distance between A and B = begin mathsize 12px style square root of open parentheses negative 1 minus 3 close parentheses squared plus left parenthesis negative 3 plus 4 right parenthesis squared end root space equals space square root of 17 end style
distance between C and D = begin mathsize 12px style square root of open parentheses negative 6 minus x close parentheses squared plus open parentheses 2 minus y close parentheses squared end root space equals space square root of 17 end style.............(1)
eqn.(1) can be written as, x2 + y2 + 12x - 4y+23 = 0 .....................(2)
 
distance between B and C = begin mathsize 12px style square root of open parentheses negative 1 plus 6 close parentheses squared plus open parentheses negative 3 minus 2 close parentheses squared end root space equals space square root of 50 end style
distance between A and D = begin mathsize 12px style square root of open parentheses 3 minus x close parentheses squared plus open parentheses negative 4 minus y close parentheses squared end root space equals space square root of 50 space end style.........................(3)
eqn.(3) can be written as,   x2 + y2 -6x + 8y -25 = 0 .....................(4)
 
equatipn eqn.(2) and eqn.(4), we get after simplification,  3x - 2y = -8 ....................(5)
 
slope of BA = begin mathsize 12px style fraction numerator negative 4 plus 3 over denominator 3 plus 1 end fraction space equals space fraction numerator negative 1 over denominator 4 end fraction end style
slope of CD = begin mathsize 12px style fraction numerator y minus 2 over denominator x plus 6 end fraction space equals space minus 1 fourth end style .....................(6)
Eqn.(6) can be reduced to  x+4y = -2 .................(7)
 
By solving eqns.(5) and (7), we get x = -2  and y = 1
 
Hence coordinate of D is ( -2 , 1 )
 
Area of parallelogram = 2×area of ΔBAD = [ -1×(-4-1) + 3×(1+3) -2×(-3+4) ] = 15 square units
Answered by Thiyagarajan K | 26 Nov, 2018, 10:58: AM

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