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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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3. Show that the points A(1,1), B(-1,-1), C(-1.732051,1.732051) are vertices of an equilateral triangle
(Previous example) | 5. Determine if the points A(0,0), B(2,0), C(-4,0), D(-2,0) are collinear points
(Next example) |
4. Show that the points A(7,10), B(-2,5), C(3,-4) are vertices of an isosceles triangle
1. Show that the points `A(7,10), B(-2,5), C(3,-4)` are vertices of an isosceles triangle
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 Triangle, in which any two sides are equal, is called an isosceles triangle
The given points are `A(7,10),B(-2,5),C(3,-4)`
`AB=sqrt((-2-7)^2+(5-10)^2)`
`=sqrt((-9)^2+(-5)^2)`
`=sqrt(81+25)`
`=sqrt(106)`
`:. AB=sqrt(106)`
`BC=sqrt((3+2)^2+(-4-5)^2)`
`=sqrt((5)^2+(-9)^2)`
`=sqrt(25+81)`
`=sqrt(106)`
`:. BC=sqrt(106)`
`AC=sqrt((3-7)^2+(-4-10)^2)`
`=sqrt((-4)^2+(-14)^2)`
`=sqrt(16+196)`
`=sqrt(212)`
`:. AC=2sqrt(53)`
Here `AB=BC`
`:.` ABC is an isoceles triangle
Also `AB^2+BC^2=(sqrt(106))^2+(sqrt(106))^2=106+106=212`
and `AC^2=(2sqrt(53))^2=212`
`:. AB^2+BC^2=AC^2` and `/_B=90^circ`
`:.` ABC is a right angle triangle
Hence, ABC is an isoceles right angle triangle
2. Show that the points `A(5,-2), B(6,4), C(7,-2)` are vertices of an isosceles triangle
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 Triangle, in which any two sides are equal, is called an isosceles triangle
The given points are `A(5,-2),B(6,4),C(7,-2)`
`AB=sqrt((6-5)^2+(4+2)^2)`
`=sqrt((1)^2+(6)^2)`
`=sqrt(1+36)`
`=sqrt(37)`
`:. AB=sqrt(37)`
`BC=sqrt((7-6)^2+(-2-4)^2)`
`=sqrt((1)^2+(-6)^2)`
`=sqrt(1+36)`
`=sqrt(37)`
`:. BC=sqrt(37)`
`AC=sqrt((7-5)^2+(-2+2)^2)`
`=sqrt((2)^2+(0)^2)`
`=sqrt(4+0)`
`=sqrt(4)`
`:. AC=2`
Here `AB=BC`
`:.` ABC is an isoceles triangle
3. Show that the points `A(3,0), B(6,4), C(-1,3)` are vertices of an isosceles triangle
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 Triangle, in which any two sides are equal, is called an isosceles triangle
The given points are `A(3,0),B(6,4),C(-1,3)`
`AB=sqrt((6-3)^2+(4-0)^2)`
`=sqrt((3)^2+(4)^2)`
`=sqrt(9+16)`
`=sqrt(25)`
`:. AB=5`
`BC=sqrt((-1-6)^2+(3-4)^2)`
`=sqrt((-7)^2+(-1)^2)`
`=sqrt(49+1)`
`=sqrt(50)`
`:. BC=5sqrt(2)`
`AC=sqrt((-1-3)^2+(3-0)^2)`
`=sqrt((-4)^2+(3)^2)`
`=sqrt(16+9)`
`=sqrt(25)`
`:. AC=5`
Here `AB=AC`
`:.` ABC is an isoceles triangle
Also `AB^2+AC^2=(5)^2+(5)^2=25+25=50`
and `BC^2=(5sqrt(2))^2=50`
`:. AB^2+AC^2=BC^2` and `/_A=90^circ`
`:.` ABC is a right angle triangle
Hence, ABC is an isoceles right angle triangle
4. Show that the points `A(-1,4), B(-3,-6), C(3,-2)` are vertices of an isosceles triangle
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 Triangle, in which any two sides are equal, is called an isosceles triangle
The given points are `A(-1,4),B(-3,-6),C(3,-2)`
`AB=sqrt((-3+1)^2+(-6-4)^2)`
`=sqrt((-2)^2+(-10)^2)`
`=sqrt(4+100)`
`=sqrt(104)`
`:. AB=2sqrt(26)`
`BC=sqrt((3+3)^2+(-2+6)^2)`
`=sqrt((6)^2+(4)^2)`
`=sqrt(36+16)`
`=sqrt(52)`
`:. BC=2sqrt(13)`
`AC=sqrt((3+1)^2+(-2-4)^2)`
`=sqrt((4)^2+(-6)^2)`
`=sqrt(16+36)`
`=sqrt(52)`
`:. AC=2sqrt(13)`
Here `BC=AC`
`:.` ABC is an isoceles triangle
Also `BC^2+AC^2=(2sqrt(13))^2+(2sqrt(13))^2=52+52=104`
and `AB^2=(2sqrt(26))^2=104`
`:. BC^2+AC^2=AB^2` and `/_C=90^circ`
`:.` ABC is a right angle triangle
Hence, ABC is an isoceles right angle triangle
5. Show that the points `A(2,2), B(-2,4), C(2,6)` are vertices of an isosceles triangle
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 Triangle, in which any two sides are equal, is called an isosceles triangle
The given points are `A(2,2),B(-2,4),C(2,6)`
`AB=sqrt((-2-2)^2+(4-2)^2)`
`=sqrt((-4)^2+(2)^2)`
`=sqrt(16+4)`
`=sqrt(20)`
`:. AB=2sqrt(5)`
`BC=sqrt((2+2)^2+(6-4)^2)`
`=sqrt((4)^2+(2)^2)`
`=sqrt(16+4)`
`=sqrt(20)`
`:. BC=2sqrt(5)`
`AC=sqrt((2-2)^2+(6-2)^2)`
`=sqrt((0)^2+(4)^2)`
`=sqrt(0+16)`
`=sqrt(16)`
`:. AC=4`
Here `AB=BC`
`:.` ABC is an isoceles triangle
This material is intended as a summary. Use your textbook for detail explanation.
Any bug, improvement, feedback then
3. Show that the points A(1,1), B(-1,-1), C(-1.732051,1.732051) are vertices of an equilateral triangle
(Previous example) | 5. Determine if the points A(0,0), B(2,0), C(-4,0), D(-2,0) are collinear points
(Next example) |
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