Show that the points A(1, –3), B(13, 9), C(10, 12)
and D(–2, 0) are vertices of a rectangle.
Answers
In a rectangle opposite sides are equal diagonals are equal
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The given points are vertices of a rectangle.
Explanation:
The given points are A(1,-3) B(13, 9), C(10, 12) and D(–2, 0)
AB = √(13-1)² + (9 - (-3))
= √ (12)² + (12)²
= √ 144 + 144
= √288
= 12√2 units
BC = √(10 - 13)² + (12 - 9)²
= √(-3)² + (3)²
= √9 + 9
= √18
= 3√2 units
CD = √ ((-2) - 10)² + (0 - 12)²
= √(-12)² + (-12)²
= √144 + 144
= √288
= 12√2 units
DA = √((-2) - 1)² + (0 - (-3)²)
= √(-3)² + (3)²
= √9 + 9
= √18
= 3√2 units
Thus AB=CD=12√2units
BC = DA = 3√2 units
Also,
AC = √(10-1)² + (12-(-3))²
= √(9)² + (15)²
= √81 + 225
= √306
= 3√34 units.
BD = √((-2) - 13)² + (0 - 9)²
= √(-15)² + (-9)²
= √225 + 81
= √306
= 3√34 units
Also,
Diagonal AC = Diagonal BD
Hence, the given points from a rectangle.
The given points are vertices of a rectangle.
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