Question

Prove that the points (3, 0) (4, 5) (-1, 4) and (-2,-1) taken in order form a rhombus

Answer 1

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Answer 2

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ABCD is a rhombus.

Let these points are A(3,0) , B(4,5) , C(-1,4) , D(-2,-1).

Using distance formula,

AB² = (X2-X1)² + (Y2-Y1)²

AB² = (4-3)² + (5-0)²

AB² = 1 + 25

AB = ✓26 units.

CD² = (-2+1)² + (-1-4)²

CD² = (-1)² + (5)²

CD² = 26

CD² = ✓26 units.

AD² = (-2+1)² + (-1-4)²

AD² = (-1)² + 25

AD² = 26

AD = ✓26

And,

BC² = (-1-4)² + (4-5)²

BC² = 25+1

BC² = 26

BC = ✓26 units

Here,

OPPOSITE SIDES ARE EQUAL IT MEANS IT MAY BE A SQUARE.. SO WE HAVE TO PROOF THAT IT IS A RHOMBUS NOT A SQUARE..

AS WE ALL KNOW THAT THE OPPOSITE SIDES OF SQUARE ARE EQUAL AND DIAGONALS ARE ALSO EQUAL.. BUT IN RHOMBUS OPPOSITE SIDES ARE EQUAL BUT DIAGONALS ARE NOT EQUAL..

IF IT'S DIAGONALS ARE NOT EQUAL THEN IT WILL BE A RHOMBUS.

AC AND BD ARE THE DIAGONALS OF RHOMBUS.

USING DISTANCE FORMULA,

AC² = (-1-3)² + (4-0)²

AC² = 16+16

AC² = 32

AC= ✓32 UNITS

AND,

BC² = (-2-4)² + (-1-5)²

BC² = 64+36

BC = ✓100 = 10 UNITS.

DIAGONALS ARE NOT EQUAL SO IT IS A RHOMBUS... AND THESE POINTS ARE FORMING A RHOMBUS...

HOPE IT WILL HELP YOU.... :-)