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2. Points are Collinear or Triangle or Quadrilateral form example
( Enter your problem )
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- A(-1,-1), B(1,5), C(2,8), Find points are collinear points or triangle
- 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. Determine if the points A(0,0), B(2,0), C(-4,0), D(-2,0) are collinear points
(Previous example) | 8. Show that the points A(-4,-1), B(-2,-4), C(4,0), D(2,3) are vertices of a rectangle
(Next example) |
7. 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
6. Determine if the points A(0,0), B(2,0), C(-4,0), D(-2,0) are collinear points
(Previous example) | 8. 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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