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2. Points are Collinear or Triangle or Quadrilateral form example ( Enter your problem )
( Enter your problem )
  1. Determine if the points A(1,5), B(2,3), C(-2,-11) are collinear points
  2. Show that the points A(-3,0), B(1,-3), C(4,1) are vertices of a right angle triangle
  3. Show that the points A(1,1), B(-1,-1), C(-1.732051,1.732051) are vertices of an equilateral triangle
  4. Show that the points A(7,10), B(-2,5), C(3,-4) are vertices of an isosceles triangle
  5. Determine if the points A(0,0), B(2,0), C(-4,0), D(-2,0) are collinear points
  6. Show that the points A(1,2), B(5,4), C(3,8), D(-1,6) are vertices of a square
  7. Show that the points A(-4,-1), B(-2,-4), C(4,0), D(2,3) are vertices of a rectangle
  8. Show that the points A(3,0), B(4,5), C(-1,4), D(-2,-1) are vertices of a rhombus
  9. Show that the points A(-3,-2), B(5,-2), C(9,3), D(1,3) are vertices of a parallelogram
 		Other related methods
  1. Distance, Slope of two points
  2. Points are Collinear or Triangle or Quadrilateral form
  3. Find Ratio of line joining AB and is divided by P
  4. Find Midpoint or Trisection points or equidistant points on X-Y axis
  5. Find Centroid, Circumcenter, Area of a triangle
  6. Find the equation of a line using slope, point, X-intercept, Y-intercept
  7. Find Slope, X-intercept, Y-intercept of a line
  8. Find the equation of a line passing through point of intersection of two lines and slope or a point
  9. Find the equation of a line passing through a point and parallel or perpendicular to Line-2 or point-2 and point-3
  10. Find the equation of a line passing through point of intersection of Line-1, Line-2 and parallel or perpendicular to Line-3
  11. For two lines, find Angle, intersection point and determine if parallel or perpendicular lines
  12. Reflection of points about x-axis, y-axis, origin
5. Determine if the points A(0,0), B(2,0), C(-4,0), D(-2,0) are collinear points
(Previous example)		7. Show that the points A(-4,-1), B(-2,-4), C(4,0), D(2,3) are vertices of a rectangle
(Next example)

6. Show that the points A(1,2), B(5,4), C(3,8), D(-1,6) are vertices of a square


1. Show that the points `A(1,2), B(5,4), C(3,8), D(-1,6)` are vertices of a square

Solution:
We know that the distance between the two points `(x_1,y_1)` and `(x_2,y_2)` is
`d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)`


A quadrilateral, in which all sides are equal and also the diagonals are equal, is a square.

So, we have to prove all sides `AB=BC=CD=AD` and both diagonals `AC=BD`


The given points are `A(1,2),B(5,4),C(3,8),D(-1,6)`

Length of sides:

`AB=sqrt((5-1)^2+(4-2)^2)`

`=sqrt((4)^2+(2)^2)`

`=sqrt(16+4)`

`=sqrt(20)`

`:. AB=2sqrt(5)`


`BC=sqrt((3-5)^2+(8-4)^2)`

`=sqrt((-2)^2+(4)^2)`

`=sqrt(4+16)`

`=sqrt(20)`

`:. BC=2sqrt(5)`


`CD=sqrt((-1-3)^2+(6-8)^2)`

`=sqrt((-4)^2+(-2)^2)`

`=sqrt(16+4)`

`=sqrt(20)`

`:. CD=2sqrt(5)`


`AD=sqrt((-1-1)^2+(6-2)^2)`

`=sqrt((-2)^2+(4)^2)`

`=sqrt(4+16)`

`=sqrt(20)`

`:. AD=2sqrt(5)`


Length of diagonals:

`AC=sqrt((3-1)^2+(8-2)^2)`

`=sqrt((2)^2+(6)^2)`

`=sqrt(4+36)`

`=sqrt(40)`

`:. AC=2sqrt(10)`


`BD=sqrt((-1-5)^2+(6-4)^2)`

`=sqrt((-6)^2+(2)^2)`

`=sqrt(36+4)`

`=sqrt(40)`

`:. BD=2sqrt(10)`


Here, all sides `AB=BC=CD=AD`

and both diagonals `AC=BD`

Since, all the sides are equal and both the diagonals are equal
Hence, ABCD is a square


2. Show that the points `A(2,3), B(-2,2), C(-1,-2), D(3,-1)` are vertices of a square

Solution:
We know that the distance between the two points `(x_1,y_1)` and `(x_2,y_2)` is
`d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)`


A quadrilateral, in which all sides are equal and also the diagonals are equal, is a square.

So, we have to prove all sides `AB=BC=CD=AD` and both diagonals `AC=BD`


The given points are `A(2,3),B(-2,2),C(-1,-2),D(3,-1)`

Length of sides:

`AB=sqrt((-2-2)^2+(2-3)^2)`

`=sqrt((-4)^2+(-1)^2)`

`=sqrt(16+1)`

`=sqrt(17)`

`:. AB=sqrt(17)`


`BC=sqrt((-1+2)^2+(-2-2)^2)`

`=sqrt((1)^2+(-4)^2)`

`=sqrt(1+16)`

`=sqrt(17)`

`:. BC=sqrt(17)`


`CD=sqrt((3+1)^2+(-1+2)^2)`

`=sqrt((4)^2+(1)^2)`

`=sqrt(16+1)`

`=sqrt(17)`

`:. CD=sqrt(17)`


`AD=sqrt((3-2)^2+(-1-3)^2)`

`=sqrt((1)^2+(-4)^2)`

`=sqrt(1+16)`

`=sqrt(17)`

`:. AD=sqrt(17)`


Length of diagonals:

`AC=sqrt((-1-2)^2+(-2-3)^2)`

`=sqrt((-3)^2+(-5)^2)`

`=sqrt(9+25)`

`=sqrt(34)`

`:. AC=sqrt(34)`


`BD=sqrt((3+2)^2+(-1-2)^2)`

`=sqrt((5)^2+(-3)^2)`

`=sqrt(25+9)`

`=sqrt(34)`

`:. BD=sqrt(34)`


Here, all sides `AB=BC=CD=AD`

and both diagonals `AC=BD`

Since, all the sides are equal and both the diagonals are equal
Hence, ABCD is a square


3. Show that the points `A(1,7), B(4,2), C(-1,-1), D(-4,4)` are vertices of a square

Solution:
We know that the distance between the two points `(x_1,y_1)` and `(x_2,y_2)` is
`d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)`


A quadrilateral, in which all sides are equal and also the diagonals are equal, is a square.

So, we have to prove all sides `AB=BC=CD=AD` and both diagonals `AC=BD`


The given points are `A(1,7),B(4,2),C(-1,-1),D(-4,4)`

Length of sides:

`AB=sqrt((4-1)^2+(2-7)^2)`

`=sqrt((3)^2+(-5)^2)`

`=sqrt(9+25)`

`=sqrt(34)`

`:. AB=sqrt(34)`


`BC=sqrt((-1-4)^2+(-1-2)^2)`

`=sqrt((-5)^2+(-3)^2)`

`=sqrt(25+9)`

`=sqrt(34)`

`:. BC=sqrt(34)`


`CD=sqrt((-4+1)^2+(4+1)^2)`

`=sqrt((-3)^2+(5)^2)`

`=sqrt(9+25)`

`=sqrt(34)`

`:. CD=sqrt(34)`


`AD=sqrt((-4-1)^2+(4-7)^2)`

`=sqrt((-5)^2+(-3)^2)`

`=sqrt(25+9)`

`=sqrt(34)`

`:. AD=sqrt(34)`


Length of diagonals:

`AC=sqrt((-1-1)^2+(-1-7)^2)`

`=sqrt((-2)^2+(-8)^2)`

`=sqrt(4+64)`

`=sqrt(68)`

`:. AC=2sqrt(17)`


`BD=sqrt((-4-4)^2+(4-2)^2)`

`=sqrt((-8)^2+(2)^2)`

`=sqrt(64+4)`

`=sqrt(68)`

`:. BD=2sqrt(17)`


Here, all sides `AB=BC=CD=AD`

and both diagonals `AC=BD`

Since, all the sides are equal and both the diagonals are equal
Hence, ABCD is a square


4. Show that the points `A(3,2), B(0,5), C(-3,2), D(0,-1)` are vertices of a square

Solution:
We know that the distance between the two points `(x_1,y_1)` and `(x_2,y_2)` is
`d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)`


A quadrilateral, in which all sides are equal and also the diagonals are equal, is a square.

So, we have to prove all sides `AB=BC=CD=AD` and both diagonals `AC=BD`


The given points are `A(3,2),B(0,5),C(-3,2),D(0,-1)`

Length of sides:

`AB=sqrt((0-3)^2+(5-2)^2)`

`=sqrt((-3)^2+(3)^2)`

`=sqrt(9+9)`

`=sqrt(18)`

`:. AB=3sqrt(2)`


`BC=sqrt((-3-0)^2+(2-5)^2)`

`=sqrt((-3)^2+(-3)^2)`

`=sqrt(9+9)`

`=sqrt(18)`

`:. BC=3sqrt(2)`


`CD=sqrt((0+3)^2+(-1-2)^2)`

`=sqrt((3)^2+(-3)^2)`

`=sqrt(9+9)`

`=sqrt(18)`

`:. CD=3sqrt(2)`


`AD=sqrt((0-3)^2+(-1-2)^2)`

`=sqrt((-3)^2+(-3)^2)`

`=sqrt(9+9)`

`=sqrt(18)`

`:. AD=3sqrt(2)`


Length of diagonals:

`AC=sqrt((-3-3)^2+(2-2)^2)`

`=sqrt((-6)^2+(0)^2)`

`=sqrt(36+0)`

`=sqrt(36)`

`:. AC=6`


`BD=sqrt((0-0)^2+(-1-5)^2)`

`=sqrt((0)^2+(-6)^2)`

`=sqrt(0+36)`

`=sqrt(36)`

`:. BD=6`


Here, all sides `AB=BC=CD=AD`

and both diagonals `AC=BD`

Since, all the sides are equal and both the diagonals are equal
Hence, ABCD is a square


5. Show that the points `A(5,6), B(1,5), C(2,1), D(6,2)` are vertices of a square

Solution:
We know that the distance between the two points `(x_1,y_1)` and `(x_2,y_2)` is
`d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)`


A quadrilateral, in which all sides are equal and also the diagonals are equal, is a square.

So, we have to prove all sides `AB=BC=CD=AD` and both diagonals `AC=BD`


The given points are `A(5,6),B(1,5),C(2,1),D(6,2)`

Length of sides:

`AB=sqrt((1-5)^2+(5-6)^2)`

`=sqrt((-4)^2+(-1)^2)`

`=sqrt(16+1)`

`=sqrt(17)`

`:. AB=sqrt(17)`


`BC=sqrt((2-1)^2+(1-5)^2)`

`=sqrt((1)^2+(-4)^2)`

`=sqrt(1+16)`

`=sqrt(17)`

`:. BC=sqrt(17)`


`CD=sqrt((6-2)^2+(2-1)^2)`

`=sqrt((4)^2+(1)^2)`

`=sqrt(16+1)`

`=sqrt(17)`

`:. CD=sqrt(17)`


`AD=sqrt((6-5)^2+(2-6)^2)`

`=sqrt((1)^2+(-4)^2)`

`=sqrt(1+16)`

`=sqrt(17)`

`:. AD=sqrt(17)`


Length of diagonals:

`AC=sqrt((2-5)^2+(1-6)^2)`

`=sqrt((-3)^2+(-5)^2)`

`=sqrt(9+25)`

`=sqrt(34)`

`:. AC=sqrt(34)`


`BD=sqrt((6-1)^2+(2-5)^2)`

`=sqrt((5)^2+(-3)^2)`

`=sqrt(25+9)`

`=sqrt(34)`

`:. BD=sqrt(34)`


Here, all sides `AB=BC=CD=AD`

and both diagonals `AC=BD`

Since, all the sides are equal and both the diagonals are equal
Hence, ABCD is a square




This material is intended as a summary. Use your textbook for detail explanation.
Any bug, improvement, feedback then Submit Here


5. Determine if the points A(0,0), B(2,0), C(-4,0), D(-2,0) are collinear points
(Previous example)		7. Show that the points A(-4,-1), B(-2,-4), C(4,0), D(2,3) are vertices of a rectangle
(Next example)


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