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Asked by yashimadhavi65 | 06 Jun, 2020, 11:19: AM
answered-by-expert Expert Answer
Q. Prove that the points A(0, -1), B(-2, 3), C(6, 7) and D(8, 3) are the vertices of rectangles ABCD.
Solution:
ABCD are vertices of a rectangle if their opposite sides are equal and adjacent sides are perpendicular.
A B equals square root of open parentheses 0 plus 2 close parentheses squared plus open parentheses negative 1 minus 3 close parentheses squared end root equals square root of 4 plus 16 end root equals square root of 20 equals 2 square root of 5
B C equals square root of open parentheses negative 2 minus 6 close parentheses squared plus open parentheses 3 minus 7 close parentheses squared end root equals square root of 64 plus 16 end root equals square root of 80 equals 4 square root of 5
C D equals square root of open parentheses 6 minus 8 close parentheses squared plus open parentheses 7 minus 3 close parentheses squared end root equals square root of 4 plus 16 end root equals square root of 20 equals 2 square root of 5
A D equals square root of open parentheses 0 minus 8 close parentheses squared plus open parentheses negative 1 minus 3 close parentheses squared end root equals square root of 64 plus 16 end root equals square root of 80 equals 4 square root of 5
S l o p e space o f space A B equals fraction numerator 3 plus 1 over denominator negative 2 plus 0 end fraction equals fraction numerator 4 over denominator negative 2 end fraction equals negative 2
S l o p e space o f space B C equals fraction numerator 7 minus 3 over denominator 6 plus 2 end fraction equals 4 over 8 equals 1 half
S l o p e space o f space C D equals fraction numerator 3 minus 7 over denominator 8 minus 6 end fraction equals fraction numerator negative 4 over denominator 2 end fraction equals negative 2
S l o p e space o f space A D equals fraction numerator 3 plus 1 over denominator 8 minus 0 end fraction equals 4 over 8 equals 1 half
S l o p e space o f space A B space cross times space S l o p e space o f space B C space equals negative 2 cross times 1 half equals negative 1
S l o p e space o f space B C space cross times space S l o p e space o f space C D space equals 1 half cross times left parenthesis negative 2 right parenthesis equals negative 1
S l o p e space o f space C D space cross times space S l o p e space o f space A D space equals negative 2 cross times 1 half equals negative 1
S l o p e space o f space A D space cross times space S l o p e space o f space A B space equals 1 half cross times open parentheses negative 2 close parentheses equals negative 1
T h u s comma space a l l space t h e space a n g l e s space a r e space r i g h t space a n g l e s space a n d space o p p o s i t e space s i d e s space a r e space e q u a l space a n d space p a r a l l e l.
H e n c e comma space A B C D thin space i s space a space r e c tan g l e.
 
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