Advertisements
Advertisements
प्रश्न
Show that A (−3, 2), B (−5, −5), C (2,−3), and D (4, 4) are the vertices of a rhombus.
Advertisements
उत्तर
Let A (−3, 2); B (−5,−5); C (2,−3) and D (4, 4) be the vertices of a quadrilateral. We have to prove that the quadrilateral ABCD is a rhombus.
So we should find the lengths of sides of quadrilateral ABCD.
`AB = sqrt((-5 + 3 )^2 + (-5-2)^2)`
`= sqrt(4 + 4 9)`
`= sqrt(53) `
`BC = sqrt((2 + 5 )^2 + (-3+ 5)^2)`
`= sqrt(4 + 4 9)`
`= sqrt(53) `
`CD = sqrt(( 4 - 2 )^2 + (4 +3)^2)`
`= sqrt(4 + 4 9)`
`= sqrt(53) `
`AD= sqrt((4 + 3 )^2 + (4-2)^2)`
`= sqrt(4 + 4 9)`
`= sqrt(53) `
All the sides of quadrilateral are equal. Hence ABCD is a rhombus.
APPEARS IN
संबंधित प्रश्न
(Street Plan): A city has two main roads which cross each other at the centre of the city. These two roads are along the North-South direction and East-West direction.
All the other streets of the city run parallel to these roads and are 200 m apart. There are 5 streets in each direction. Using 1cm = 200 m, draw a model of the city on your notebook. Represent the roads/streets by single lines.
There are many cross- streets in your model. A particular cross-street is made by two streets, one running in the North - South direction and another in the East - West direction. Each cross street is referred to in the following manner : If the 2nd street running in the North - South direction and 5th in the East - West direction meet at some crossing, then we will call this cross-street (2, 5). Using this convention, find:
- how many cross - streets can be referred to as (4, 3).
- how many cross - streets can be referred to as (3, 4).
Name the quadrilateral formed, if any, by the following points, and given reasons for your answers:
A(-1,-2) B(1, 0), C (-1, 2), D(-3, 0)
Find the coordinates of the point which divides the line segment joining (−1,3) and (4, −7) internally in the ratio 3 : 4
Find the ratio in which the point (2, y) divides the line segment joining the points A (-2,2) and B (3, 7). Also, find the value of y.
Prove that (4, 3), (6, 4) (5, 6) and (3, 5) are the angular points of a square.
Find the coordinates of the points which divide the line segment joining the points (-4, 0) and (0, 6) in four equal parts.
Determine the ratio in which the point P (m, 6) divides the join of A(−4, 3) and B(2, 8). Also, find the value of m.
In what ratio does the point P(2,5) divide the join of A (8,2) and B(-6, 9)?
If the vertices of ΔABC be A(1, -3) B(4, p) and C(-9, 7) and its area is 15 square units, find the values of p
Find the coordinates of the points of trisection of the line segment joining the points (3, –2) and (–3, –4) ?
Mark the correct alternative in each of the following:
The point of intersect of the coordinate axes is
The abscissa of a point is positive in the
In \[∆\] ABC , the coordinates of vertex A are (0, - 1) and D (1,0) and E(0,10) respectively the mid-points of the sides AB and AC . If F is the mid-points of the side BC , find the area of \[∆\] DEF.
If the vertices of a triangle are (1, −3), (4, p) and (−9, 7) and its area is 15 sq. units, find the value(s) of p.
If three points (0, 0), \[\left( 3, \sqrt{3} \right)\] and (3, λ) form an equilateral triangle, then λ =
If (−1, 2), (2, −1) and (3, 1) are any three vertices of a parallelogram, then
If Points (1, 2) (−5, 6) and (a, −2) are collinear, then a =
Find the point on the y-axis which is equidistant from the points (S, - 2) and (- 3, 2).
Find the point on the y-axis which is equidistant from the points (5, −2) and (−3, 2).
In which quadrant, does the abscissa, and ordinate of a point have the same sign?
