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2. Points are Collinear or Triangle or Quadrilateral form example
( Enter your problem )
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- Determine if the points A(1,5), B(2,3), C(-2,-11) are collinear points
- Show that the points A(-3,0), B(1,-3), C(4,1) are vertices of a right angle triangle
- Show that the points A(1,1), B(-1,-1), C(-1.732051,1.732051) are vertices of an equilateral triangle
- Show that the points A(7,10), B(-2,5), C(3,-4) are vertices of an isosceles triangle
- Determine if the points A(0,0), B(2,0), C(-4,0), D(-2,0) are collinear points
- Show that the points A(1,2), B(5,4), C(3,8), D(-1,6) are vertices of a square
- Show that the points A(-4,-1), B(-2,-4), C(4,0), D(2,3) are vertices of a rectangle
- Show that the points A(3,0), B(4,5), C(-1,4), D(-2,-1) are vertices of a rhombus
- Show that the points A(-3,-2), B(5,-2), C(9,3), D(1,3) are vertices of a parallelogram
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Other related methods
- Distance, Slope of two points
- Points are Collinear or Triangle or Quadrilateral form
- Find Ratio of line joining AB and is divided by P
- Find Midpoint or Trisection points or equidistant points on X-Y axis
- Find Centroid, Circumcenter, Area of a triangle
- Find the equation of a line using slope, point, X-intercept, Y-intercept
- Find Slope, X-intercept, Y-intercept of a line
- Find the equation of a line passing through point of intersection of two lines and slope or a point
- Find the equation of a line passing through a point and parallel or perpendicular to Line-2 or point-2 and point-3
- Find the equation of a line passing through point of intersection of Line-1, Line-2 and parallel or perpendicular to Line-3
- For two lines, find Angle, intersection point and determine if parallel or perpendicular lines
- Reflection of points about x-axis, y-axis, origin
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6. Show that the points A(1,2), B(5,4), C(3,8), D(-1,6) are vertices of a square
(Previous example) | 8. Show that the points A(3,0), B(4,5), C(-1,4), D(-2,-1) are vertices of a rhombus
(Next example) |
7. Show that the points A(-4,-1), B(-2,-4), C(4,0), D(2,3) are vertices of a rectangle
1. Show that the points `A(-4,-1), B(-2,-4), C(4,0), D(2,3)` are vertices of a rectangle
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 opposites sides are equal and also the diagonals are equal, is a rectangle.
So, we have to prove opposite sides `AB=CD` and `BC=AD` and both diagonals `AC=BD`
The given points are `A(-4,-1),B(-2,-4),C(4,0),D(2,3)`
Length of sides:
`AB=sqrt((-2+4)^2+(-4+1)^2)`
`=sqrt((2)^2+(-3)^2)`
`=sqrt(4+9)`
`=sqrt(13)`
`:. AB=sqrt(13)`
`CD=sqrt((2-4)^2+(3-0)^2)`
`=sqrt((-2)^2+(3)^2)`
`=sqrt(4+9)`
`=sqrt(13)`
`:. CD=sqrt(13)`
`BC=sqrt((4+2)^2+(0+4)^2)`
`=sqrt((6)^2+(4)^2)`
`=sqrt(36+16)`
`=sqrt(52)`
`:. BC=2sqrt(13)`
`AD=sqrt((2+4)^2+(3+1)^2)`
`=sqrt((6)^2+(4)^2)`
`=sqrt(36+16)`
`=sqrt(52)`
`:. AD=2sqrt(13)`
Length of diagonals:
`AC=sqrt((4+4)^2+(0+1)^2)`
`=sqrt((8)^2+(1)^2)`
`=sqrt(64+1)`
`=sqrt(65)`
`:. AC=sqrt(65)`
`BD=sqrt((2+2)^2+(3+4)^2)`
`=sqrt((4)^2+(7)^2)`
`=sqrt(16+49)`
`=sqrt(65)`
`:. BD=sqrt(65)`
Here opposite sides `AB=CD` and `BC=AD`
and both diagonals `AC=BD`
Since, all the opposite sides are equal and both the diagonals are equal
Hence, ABCD is a rectangle
2. Show that the points `A(2,-2), B(8,4), C(5,7), D(-1,1)` are vertices of a rectangle
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 opposites sides are equal and also the diagonals are equal, is a rectangle.
So, we have to prove opposite sides `AB=CD` and `BC=AD` and both diagonals `AC=BD`
The given points are `A(2,-2),B(8,4),C(5,7),D(-1,1)`
Length of sides:
`AB=sqrt((8-2)^2+(4+2)^2)`
`=sqrt((6)^2+(6)^2)`
`=sqrt(36+36)`
`=sqrt(72)`
`:. AB=6sqrt(2)`
`CD=sqrt((-1-5)^2+(1-7)^2)`
`=sqrt((-6)^2+(-6)^2)`
`=sqrt(36+36)`
`=sqrt(72)`
`:. CD=6sqrt(2)`
`BC=sqrt((5-8)^2+(7-4)^2)`
`=sqrt((-3)^2+(3)^2)`
`=sqrt(9+9)`
`=sqrt(18)`
`:. BC=3sqrt(2)`
`AD=sqrt((-1-2)^2+(1+2)^2)`
`=sqrt((-3)^2+(3)^2)`
`=sqrt(9+9)`
`=sqrt(18)`
`:. AD=3sqrt(2)`
Length of diagonals:
`AC=sqrt((5-2)^2+(7+2)^2)`
`=sqrt((3)^2+(9)^2)`
`=sqrt(9+81)`
`=sqrt(90)`
`:. AC=3sqrt(10)`
`BD=sqrt((-1-8)^2+(1-4)^2)`
`=sqrt((-9)^2+(-3)^2)`
`=sqrt(81+9)`
`=sqrt(90)`
`:. BD=3sqrt(10)`
Here opposite sides `AB=CD` and `BC=AD`
and both diagonals `AC=BD`
Since, all the opposite sides are equal and both the diagonals are equal
Hence, ABCD is a rectangle
3. Show that the points `A(0,-1), B(-2,3), C(6,7), D(8,3)` are vertices of a rectangle
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 opposites sides are equal and also the diagonals are equal, is a rectangle.
So, we have to prove opposite sides `AB=CD` and `BC=AD` and both diagonals `AC=BD`
The given points are `A(0,-1),B(-2,3),C(6,7),D(8,3)`
Length of sides:
`AB=sqrt((-2-0)^2+(3+1)^2)`
`=sqrt((-2)^2+(4)^2)`
`=sqrt(4+16)`
`=sqrt(20)`
`:. AB=2sqrt(5)`
`CD=sqrt((8-6)^2+(3-7)^2)`
`=sqrt((2)^2+(-4)^2)`
`=sqrt(4+16)`
`=sqrt(20)`
`:. CD=2sqrt(5)`
`BC=sqrt((6+2)^2+(7-3)^2)`
`=sqrt((8)^2+(4)^2)`
`=sqrt(64+16)`
`=sqrt(80)`
`:. BC=4sqrt(5)`
`AD=sqrt((8-0)^2+(3+1)^2)`
`=sqrt((8)^2+(4)^2)`
`=sqrt(64+16)`
`=sqrt(80)`
`:. AD=4sqrt(5)`
Length of diagonals:
`AC=sqrt((6-0)^2+(7+1)^2)`
`=sqrt((6)^2+(8)^2)`
`=sqrt(36+64)`
`=sqrt(100)`
`:. AC=10`
`BD=sqrt((8+2)^2+(3-3)^2)`
`=sqrt((10)^2+(0)^2)`
`=sqrt(100+0)`
`=sqrt(100)`
`:. BD=10`
Here opposite sides `AB=CD` and `BC=AD`
and both diagonals `AC=BD`
Since, all the opposite sides are equal and both the diagonals are equal
Hence, ABCD is a rectangle
4. Show that the points `A(2,-2), B(14,10), C(11,13), D(-1,1)` are vertices of a rectangle
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 opposites sides are equal and also the diagonals are equal, is a rectangle.
So, we have to prove opposite sides `AB=CD` and `BC=AD` and both diagonals `AC=BD`
The given points are `A(2,-2),B(14,10),C(11,13),D(-1,1)`
Length of sides:
`AB=sqrt((14-2)^2+(10+2)^2)`
`=sqrt((12)^2+(12)^2)`
`=sqrt(144+144)`
`=sqrt(288)`
`:. AB=12sqrt(2)`
`CD=sqrt((-1-11)^2+(1-13)^2)`
`=sqrt((-12)^2+(-12)^2)`
`=sqrt(144+144)`
`=sqrt(288)`
`:. CD=12sqrt(2)`
`BC=sqrt((11-14)^2+(13-10)^2)`
`=sqrt((-3)^2+(3)^2)`
`=sqrt(9+9)`
`=sqrt(18)`
`:. BC=3sqrt(2)`
`AD=sqrt((-1-2)^2+(1+2)^2)`
`=sqrt((-3)^2+(3)^2)`
`=sqrt(9+9)`
`=sqrt(18)`
`:. AD=3sqrt(2)`
Length of diagonals:
`AC=sqrt((11-2)^2+(13+2)^2)`
`=sqrt((9)^2+(15)^2)`
`=sqrt(81+225)`
`=sqrt(306)`
`:. AC=3sqrt(34)`
`BD=sqrt((-1-14)^2+(1-10)^2)`
`=sqrt((-15)^2+(-9)^2)`
`=sqrt(225+81)`
`=sqrt(306)`
`:. BD=3sqrt(34)`
Here opposite sides `AB=CD` and `BC=AD`
and both diagonals `AC=BD`
Since, all the opposite sides are equal and both the diagonals are equal
Hence, ABCD is a rectangle
5. Show that the points `A(0,-1), B(-2,3), C(6,7), D(8,3)` are vertices of a rectangle
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 opposites sides are equal and also the diagonals are equal, is a rectangle.
So, we have to prove opposite sides `AB=CD` and `BC=AD` and both diagonals `AC=BD`
The given points are `A(0,-1),B(-2,3),C(6,7),D(8,3)`
Length of sides:
`AB=sqrt((-2-0)^2+(3+1)^2)`
`=sqrt((-2)^2+(4)^2)`
`=sqrt(4+16)`
`=sqrt(20)`
`:. AB=2sqrt(5)`
`CD=sqrt((8-6)^2+(3-7)^2)`
`=sqrt((2)^2+(-4)^2)`
`=sqrt(4+16)`
`=sqrt(20)`
`:. CD=2sqrt(5)`
`BC=sqrt((6+2)^2+(7-3)^2)`
`=sqrt((8)^2+(4)^2)`
`=sqrt(64+16)`
`=sqrt(80)`
`:. BC=4sqrt(5)`
`AD=sqrt((8-0)^2+(3+1)^2)`
`=sqrt((8)^2+(4)^2)`
`=sqrt(64+16)`
`=sqrt(80)`
`:. AD=4sqrt(5)`
Length of diagonals:
`AC=sqrt((6-0)^2+(7+1)^2)`
`=sqrt((6)^2+(8)^2)`
`=sqrt(36+64)`
`=sqrt(100)`
`:. AC=10`
`BD=sqrt((8+2)^2+(3-3)^2)`
`=sqrt((10)^2+(0)^2)`
`=sqrt(100+0)`
`=sqrt(100)`
`:. BD=10`
Here opposite sides `AB=CD` and `BC=AD`
and both diagonals `AC=BD`
Since, all the opposite sides are equal and both the diagonals are equal
Hence, ABCD is a rectangle
This material is intended as a summary. Use your textbook for detail explanation.
Any bug, improvement, feedback then
6. Show that the points A(1,2), B(5,4), C(3,8), D(-1,6) are vertices of a square
(Previous example) | 8. Show that the points A(3,0), B(4,5), C(-1,4), D(-2,-1) are vertices of a rhombus
(Next example) |
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