Advertisements
Advertisements
प्रश्न
Name the quadrilateral formed, if any, by the following points, and given reasons for your answers:
A(4, 5) B(7, 6), C (4, 3), D(1, 2)
Advertisements
उत्तर
A (4, 5), B (7,6), C(4,3), D(1,2)
Let A, B, C and D be the four vertices of the quadrilateral ABCD.
We know the distance between two points `P(x_1,y_1)` and `Q(x_2, y_2)is given by distance formula:
`PQ = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)`
Hence
`=> AB = sqrt((7 - 4)^2 + (6 - 5)^2)`
`=> AB = sqrt((-3)^2 + (-3)^2)`
`=> AB = sqrt(9 + 1)`
`=> AB = sqrt(10)`
Similarly,
`=> BC = sqrt((4 - 7)^2 + (3 - 6)^2)`
`=> BC = sqrt((-3)^2 + (1)^2)`
`=> BC= sqrt(9 + 9)`
`=> BC = sqrt18`
Similarly
`=> CD = sqrt((1 - 4)^2 + (2 - 3)^2)`
`=> CD = sqrt((-3)^2 + (-1)^2)`
`=> CD = sqrt(9 + 1)`
`=> CD = sqrt(9 + 1)`
`=> CD = sqrt10`
Also
`=> DA = sqrt((1 - 4)^2 + (2 -5)^2)`
`=> DA = sqrt((-3)^2 + (-3)^2)`
`=> DA = sqrt(9 + 9)`
`=> DA = sqrt18`
Hence from above we see that
AB = CD and BC = DA
Hence from above we see that
AB = CD and BC = DA
Here opposite sides of the quadrilateral is equal. Hence it is a parallelogram
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).
Which point on the x-axis is equidistant from (5, 9) and (−4, 6)?
If the points p (x , y) is point equidistant from the points A (5,1)and B ( -1,5) , Prove that 3x=2y
If the point ( x,y ) is equidistant form the points ( a+b,b-a ) and (a-b ,a+b ) , prove that bx = ay
Show that the following points are the vertices of a square:
A (6,2), B(2,1), C(1,5) and D(5,6)
If (2, p) is the midpoint of the line segment joining the points A(6, -5) and B(-2,11) find the value of p.
Find the area of the triangle formed by joining the midpoints of the sides of the triangle whose vertices are A(2,1) B(4,3) and C(2,5)
Prove that the diagonals of a rectangle ABCD with vertices A(2,-1), B(5,-1) C(5,6) and D(2,6) are equal and bisect each other
If the points A (2,3), B (4,k ) and C (6,-3) are collinear, find the value of k.
A point whose abscissa is −3 and ordinate 2 lies in
If the distance between the points (4, p) and (1, 0) is 5, then p =
The area of the triangle formed by (a, b + c), (b, c + a) and (c, a + b)
If the area of the triangle formed by the points (x, 2x), (−2, 6) and (3, 1) is 5 square units , then x =
The ratio in which the x-axis divides the segment joining (3, 6) and (12, −3) is
The length of a line segment joining A (2, −3) and B is 10 units. If the abscissa of B is 10 units, then its ordinates can be
In which quadrant does the point (-4, -3) lie?
The line segment joining the points A(2, 1) and B (5, - 8) is trisected at the points P and Q such that P is nearer to A. If P also lies on the line given by 2x - y + k= 0 find the value of k.
In the above figure, seg PA, seg QB and RC are perpendicular to seg AC. From the information given in the figure, prove that: `1/x + 1/y = 1/z`
Abscissa of all the points on the x-axis is ______.
Ryan, from a very young age, was fascinated by the twinkling of stars and the vastness of space. He always dreamt of becoming an astronaut one day. So, he started to sketch his own rocket designs on the graph sheet. One such design is given below :
Based on the above, answer the following questions:
i. Find the mid-point of the segment joining F and G. (1)
ii. a. What is the distance between the points A and C? (2)
OR
b. Find the coordinates of the points which divides the line segment joining the points A and B in the ratio 1 : 3 internally. (2)
iii. What are the coordinates of the point D? (1)
