The extremities of the diagonal of a square are the
points A (1,5), B (8,8). Find the coordinates of the
other vertices
Answers
(3 , 10) & ( 6 , 3) are the coordinates of the other vertices
Step-by-step explanation:
The extremities of the diagonal of a square are the
points A (1,5), B (8,8)
Diagonal of Squares perpendicularly bisects Each other
=> Mid point of Diagonal = (1 + 8)/2 , (5 + 8)/2
= 9/2 , 13/2
Slope of Diagonal = (8 - 5)/(8 - 1) = 3/7
Slope of other diagonal = -7/3
Length of half diagonal = √(8 - 9/2)² + (8 - 13/2)²
= √(7/2)² + (3/2)²
= √(49 + 9)/4
= √58 / 2
y = mx + c
=> y = -7x/3 + c
Passes through 9/2 , 13/2
=> 13/2 = (-7/3)(9/2) + c
=> 39 = -63 + 6c
=> 6c = 102
=> c = 17
y = -7x/3 + 17
Other two vertices are ( a , b) & (c , d)
b = -7a/3 + 17
√((b - 13/2)² + (a - 9/2)²) = √58 / 2
=> (b - 13/2)² + (a - 9/2)² = 58/4
=> (2b - 13)² + (2a - 9)² = 58
=> (2(-7a/3 + 17) - 13)² + (2a - 9)² = 58
=> (-14a + 63)² + 9(2a - 9)² = 9 * 58
=> 196a² + 63² -28a * 63 + 9(4a² + 81 - 36a) = 9 * 58
=> 232a² -2088a + 4176 = 0
=> a² - 9a + 18 = 0
=> a² - 6a - 3a + 18 = 0
=> (a - 3)(a - 6) =0
=> a = 3 , 6
a = 3 , b = -7a/3 + 17 = 10
a = 6 , b = 3
(3 , 10) & ( 6 , 3)
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