A certain square is to be drawn on a rectangular coordinate system. One of its vertices must be on the origin, and the square is to have an area of 25. If all coordinates of the vertices must be integers, how many different ways this square can be drawn?
Choose one of the following answers
4
8
12
24
Answers
Given : A certain square is to be drawn on a rectangular coordinate system. One of its vertices must be on the origin, and the square is to have an area of 25.
all coordinates of the vertices must be integers,
To Find : how many different ways this square can be drawn
4
8
12
24
Solution:
area of square is 25
side length = 5
0² + 5² = 25
3² + 4² = 25
4² + 3² = 25
hence another one coordinates can be of form ( 0 , ±5) , (±5 , 0) ,
(±3 , ±4) , (±4 , ±3)
so this leads to :
( 0 , 5) , (0 , -5) , ( 5 , 0) , (-5 , 0)
( 3 , 4) , ( 3 , -4 ) , ( - 3 , 4) , ( - 3, - 4)
( 4 , 3) , ( 4 , -3 ) , ( - 4 , 3) , ( - 4, - 3)
Total 12 Square are possible
below are detailed coordinates:
( 0 , 0) , ( 5 , 0) , ( 5 , 5) , ( 0 , 5)
( 0 , 0) , ( 5 , 0) , ( 5 , -5) , ( 0 , -5)
( 0 , 0) , (- 5 , 0) , ( -5 , 5) , ( 0 , 5)
( 0 , 0) , ( -5 , 0) , ( -5 , -5) , ( 0 , -5)
( 0 , 0) , ( 3 , 4) , ( -1 , 7) , ( -4 ,3)
( 0 , 0) , ( 3 , 4) , (7 , 1) , ( 4 ,-3)
( 0 , 0) , ( -3 , -4) , ( -7 , -1) , ( -4 ,3)
( 0 , 0) , ( -3 , -4) , (1 , -7) , ( 4 ,-3)
( 0 , 0) , ( 4 , 3) , ( 1 , 7) , ( -3 ,4)
( 0 , 0) , ( 4 , 3) , (7 , -1) , ( 3 ,-4)
( 0 , 0) , ( -4 ,- 3) , ( -7 , 1) , ( -3 ,4)
( 0 , 0) , ( -4 , -3) , (-1 , -7) , ( 3 ,--4)
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