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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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7. Show that the points A(-4,-1), B(-2,-4), C(4,0), D(2,3) are vertices of a rectangle
(Previous example) | 9. Show that the points A(-3,-2), B(5,-2), C(9,3), D(1,3) are vertices of a parallelogram
(Next example) |
8. Show that the points A(3,0), B(4,5), C(-1,4), D(-2,-1) are vertices of a rhombus
1. Show that the points `A(3,0), B(4,5), C(-1,4), D(-2,-1)` are vertices of a rhombus
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, is a rhombus.
So, we have to prove all sides `AB=BC=CD=AD` and both diagonals `AC!=BD`
The given points are `A(3,0),B(4,5),C(-1,4),D(-2,-1)`
Length of sides:
`AB=sqrt((4-3)^2+(5-0)^2)`
`=sqrt((1)^2+(5)^2)`
`=sqrt(1+25)`
`=sqrt(26)`
`:. AB=sqrt(26)`
`BC=sqrt((-1-4)^2+(4-5)^2)`
`=sqrt((-5)^2+(-1)^2)`
`=sqrt(25+1)`
`=sqrt(26)`
`:. BC=sqrt(26)`
`CD=sqrt((-2+1)^2+(-1-4)^2)`
`=sqrt((-1)^2+(-5)^2)`
`=sqrt(1+25)`
`=sqrt(26)`
`:. CD=sqrt(26)`
`AD=sqrt((-2-3)^2+(-1-0)^2)`
`=sqrt((-5)^2+(-1)^2)`
`=sqrt(25+1)`
`=sqrt(26)`
`:. AD=sqrt(26)`
Length of diagonals:
`AC=sqrt((-1-3)^2+(4-0)^2)`
`=sqrt((-4)^2+(4)^2)`
`=sqrt(16+16)`
`=sqrt(32)`
`:. AC=4sqrt(2)`
`BD=sqrt((-2-4)^2+(-1-5)^2)`
`=sqrt((-6)^2+(-6)^2)`
`=sqrt(36+36)`
`=sqrt(72)`
`:. BD=6sqrt(2)`
Here all sides `AB=BC=CD=AD` and both diagonals `AC!=BD`
Since, all the sides are equal and both the diagonals are not equal
Hence, ABCD is a rhombus
Area `=1/2 xx AC xx BD`
`=1/2 xx 4sqrt(2) xx 6sqrt(2)`
`=24`
Hence, the area of the rhombus is `24` square units
2. Show that the points `A(-3,2), B(-5,-5), C(2,-3), D(4,4)` are vertices of a rhombus
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, is a rhombus.
So, we have to prove all sides `AB=BC=CD=AD` and both diagonals `AC!=BD`
The given points are `A(-3,2),B(-5,-5),C(2,-3),D(4,4)`
Length of sides:
`AB=sqrt((-5+3)^2+(-5-2)^2)`
`=sqrt((-2)^2+(-7)^2)`
`=sqrt(4+49)`
`=sqrt(53)`
`:. AB=sqrt(53)`
`BC=sqrt((2+5)^2+(-3+5)^2)`
`=sqrt((7)^2+(2)^2)`
`=sqrt(49+4)`
`=sqrt(53)`
`:. BC=sqrt(53)`
`CD=sqrt((4-2)^2+(4+3)^2)`
`=sqrt((2)^2+(7)^2)`
`=sqrt(4+49)`
`=sqrt(53)`
`:. CD=sqrt(53)`
`AD=sqrt((4+3)^2+(4-2)^2)`
`=sqrt((7)^2+(2)^2)`
`=sqrt(49+4)`
`=sqrt(53)`
`:. AD=sqrt(53)`
Length of diagonals:
`AC=sqrt((2+3)^2+(-3-2)^2)`
`=sqrt((5)^2+(-5)^2)`
`=sqrt(25+25)`
`=sqrt(50)`
`:. AC=5sqrt(2)`
`BD=sqrt((4+5)^2+(4+5)^2)`
`=sqrt((9)^2+(9)^2)`
`=sqrt(81+81)`
`=sqrt(162)`
`:. BD=9sqrt(2)`
Here all sides `AB=BC=CD=AD` and both diagonals `AC!=BD`
Since, all the sides are equal and both the diagonals are not equal
Hence, ABCD is a rhombus
Area `=1/2 xx AC xx BD`
`=1/2 xx 5sqrt(2) xx 9sqrt(2)`
`=45`
Hence, the area of the rhombus is `45` square units
3. Show that the points `A(4,-1), B(6,0), C(7,2), D(5,1)` are vertices of a rhombus
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, is a rhombus.
So, we have to prove all sides `AB=BC=CD=AD` and both diagonals `AC!=BD`
The given points are `A(4,-1),B(6,0),C(7,2),D(5,1)`
Length of sides:
`AB=sqrt((6-4)^2+(0+1)^2)`
`=sqrt((2)^2+(1)^2)`
`=sqrt(4+1)`
`=sqrt(5)`
`:. AB=sqrt(5)`
`BC=sqrt((7-6)^2+(2-0)^2)`
`=sqrt((1)^2+(2)^2)`
`=sqrt(1+4)`
`=sqrt(5)`
`:. BC=sqrt(5)`
`CD=sqrt((5-7)^2+(1-2)^2)`
`=sqrt((-2)^2+(-1)^2)`
`=sqrt(4+1)`
`=sqrt(5)`
`:. CD=sqrt(5)`
`AD=sqrt((5-4)^2+(1+1)^2)`
`=sqrt((1)^2+(2)^2)`
`=sqrt(1+4)`
`=sqrt(5)`
`:. AD=sqrt(5)`
Length of diagonals:
`AC=sqrt((7-4)^2+(2+1)^2)`
`=sqrt((3)^2+(3)^2)`
`=sqrt(9+9)`
`=sqrt(18)`
`:. AC=3sqrt(2)`
`BD=sqrt((5-6)^2+(1-0)^2)`
`=sqrt((-1)^2+(1)^2)`
`=sqrt(1+1)`
`=sqrt(2)`
`:. BD=sqrt(2)`
Here all sides `AB=BC=CD=AD` and both diagonals `AC!=BD`
Since, all the sides are equal and both the diagonals are not equal
Hence, ABCD is a rhombus
Area `=1/2 xx AC xx BD`
`=1/2 xx 3sqrt(2) xx sqrt(2)`
`=3`
Hence, the area of the rhombus is `3` square units
4. Show that the points `A(1,0), B(5,3), C(2,7), D(-2,4)` are vertices of a rhombus
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, is a rhombus.
So, we have to prove all sides `AB=BC=CD=AD` and both diagonals `AC!=BD`
The given points are `A(1,0),B(5,3),C(2,7),D(-2,4)`
Length of sides:
`AB=sqrt((5-1)^2+(3-0)^2)`
`=sqrt((4)^2+(3)^2)`
`=sqrt(16+9)`
`=sqrt(25)`
`:. AB=5`
`BC=sqrt((2-5)^2+(7-3)^2)`
`=sqrt((-3)^2+(4)^2)`
`=sqrt(9+16)`
`=sqrt(25)`
`:. BC=5`
`CD=sqrt((-2-2)^2+(4-7)^2)`
`=sqrt((-4)^2+(-3)^2)`
`=sqrt(16+9)`
`=sqrt(25)`
`:. CD=5`
`AD=sqrt((-2-1)^2+(4-0)^2)`
`=sqrt((-3)^2+(4)^2)`
`=sqrt(9+16)`
`=sqrt(25)`
`:. AD=5`
Length of diagonals:
`AC=sqrt((2-1)^2+(7-0)^2)`
`=sqrt((1)^2+(7)^2)`
`=sqrt(1+49)`
`=sqrt(50)`
`:. AC=5sqrt(2)`
`BD=sqrt((-2-5)^2+(4-3)^2)`
`=sqrt((-7)^2+(1)^2)`
`=sqrt(49+1)`
`=sqrt(50)`
`:. BD=5sqrt(2)`
Here `!=`
`:.` ABCD is not a rhombus
5. Show that the points `A(7,3), B(3,0), C(0,-4), D(4,-1)` are vertices of a rhombus
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, is a rhombus.
So, we have to prove all sides `AB=BC=CD=AD` and both diagonals `AC!=BD`
The given points are `A(7,3),B(3,0),C(0,-4),D(4,-1)`
Length of sides:
`AB=sqrt((3-7)^2+(0-3)^2)`
`=sqrt((-4)^2+(-3)^2)`
`=sqrt(16+9)`
`=sqrt(25)`
`:. AB=5`
`BC=sqrt((0-3)^2+(-4-0)^2)`
`=sqrt((-3)^2+(-4)^2)`
`=sqrt(9+16)`
`=sqrt(25)`
`:. BC=5`
`CD=sqrt((4-0)^2+(-1+4)^2)`
`=sqrt((4)^2+(3)^2)`
`=sqrt(16+9)`
`=sqrt(25)`
`:. CD=5`
`AD=sqrt((4-7)^2+(-1-3)^2)`
`=sqrt((-3)^2+(-4)^2)`
`=sqrt(9+16)`
`=sqrt(25)`
`:. AD=5`
Length of diagonals:
`AC=sqrt((0-7)^2+(-4-3)^2)`
`=sqrt((-7)^2+(-7)^2)`
`=sqrt(49+49)`
`=sqrt(98)`
`:. AC=7sqrt(2)`
`BD=sqrt((4-3)^2+(-1-0)^2)`
`=sqrt((1)^2+(-1)^2)`
`=sqrt(1+1)`
`=sqrt(2)`
`:. BD=sqrt(2)`
Here all sides `AB=BC=CD=AD` and both diagonals `AC!=BD`
Since, all the sides are equal and both the diagonals are not equal
Hence, ABCD is a rhombus
Area `=1/2 xx AC xx BD`
`=1/2 xx 7sqrt(2) xx sqrt(2)`
`=7`
Hence, the area of the rhombus is `7` square units
This material is intended as a summary. Use your textbook for detail explanation.
Any bug, improvement, feedback then
7. Show that the points A(-4,-1), B(-2,-4), C(4,0), D(2,3) are vertices of a rectangle
(Previous example) | 9. Show that the points A(-3,-2), B(5,-2), C(9,3), D(1,3) are vertices of a parallelogram
(Next example) |
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