Length is easy
length = √[(x2-x1)² + (y2-y1)²]
So the length from (-3,1) to (-1,4) is √[(-1--3)² + (4-1)²] = √[4 + 9] = √13
And the length from (-3,1) to (0,-1) is √[(0--3)² + (-1-1)²] = √[9 + 4] = √13
To prove they are perpendicular to each other just find the gradients of each side and two of the lines will have gradient m whilst the other two sides will have gradient -1/m
The gradient can be found using (y2 - y1)/(x2 - x1)
So the gradient from (-3,1) to (-1,4) is => (4-1)/(-1-1) = -3/2
And the gradient from (-3,1) to (0,-1) is => (-1-1)/(0--3) = 2/3
So just do this for the other two sides, proving their lengths are root 13, and their gradients are m, -1/m, m and -1/m.
(b) is similar. Work out the equations of the diagonals using y = mx + c, filling in the values of y and x for each pair of diagnol coordinates. Then show that the gradient, m, for one diagonal is perpendicular to the other diagnol with gradient -1/m.
(c) is just working out the length of the diagonals using the formulae i gave you in part (a). Simple enough. The values of these lines should come out as √26 since this is what pythagorus theorem suggests.
Using pythagorus theorem => a² = b² + c²
a² = (√13)² + (√13)²
a² = 13 + 13 = 26
a = √26
Length = √[(2 - 1)² + (2 - -3)²] = √[1 + 25] = √26
]]>The dashed lines are the diagonals, and they do "bisect" (cut into two equal parts) each other, and they are equal in length
You can also do it mathematically, using:
length = √ (x² + y²)
for example, the diagonals are 1 unit in one one direction and 5 in the other apart:
length = √ (1² + 5²) = √ (1 + 25) = 5.1 (approx)
]]>a) a quadrilateral is a square
b) each diagonal of the quadrilateral is the perpendicular bisector of the other diagonal
c) the diagonals of the quadrilateral are equal in length
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