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प्रश्न
The base BC of an equilateral triangle ABC lies on y-axis. The coordinates of point C are (0, -3). The origin is the midpoint of the base. Find the coordinates of the points A and B. Also, find the coordinates of another point D such that ABCD is a rhombus.
उत्तर
Let (0, y) be the coordinates of B. Then
` 0= (-3+y)/2 ⇒ y=3`
Thus, the coordinates of B are (0,3)
Here. AB = BC = AC and by symmetry the coordinates of A lies on x-axis Let the coordinates of A be (x, 0). Then
`AB= BC⇒AB^2 = BC^2`
`⇒ (x-0)^2 +(0-3)^2 = 6^2`
`⇒ x^2 = 36-9=27`
`⇒ x = +- 3 sqrt(3) `
`"If the coordinates of point A are "(3 sqrt(3),0) ."then the coordinates of D are " (-3 sqrt(3), 0).`
`"If the coordinates of point A are "(-3 sqrt(3),0) ."then the coordinates of D are " (-3 sqrt(3), 0).`
`"Hence the required coordinates are " A(3sqrt(3),0) , B(0,3) and D (-3 sqrt(3),0) or `
`A (-3sqrt(3),0) , B(0,3) and D (3sqrt(3),0).`
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).
Find the points of trisection of the line segment joining the points:
5, −6 and (−7, 5),
Find the points of trisection of the line segment joining the points:
(2, -2) and (-7, 4).
If the points A (a, -11), B (5, b), C (2, 15) and D (1, 1) are the vertices of a parallelogram ABCD, find the values of a and b.
In what ratio does the point (−4, 6) divide the line segment joining the points A(−6, 10) and B(3,−8)?
If the points p (x , y) is point equidistant from the points A (5,1)and B ( -1,5) , Prove that 3x=2y
Show that the following points are the vertices of a square:
A (6,2), B(2,1), C(1,5) and D(5,6)
Show that the points A(2,1), B(5,2), C(6,4) and D(3,3) are the angular points of a parallelogram. Is this figure a rectangle?
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 ratio in which the poin "p (3/4 , 5/12) " divides the line segment joining the points "A (1/2,3/2) and B (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 point P (m, 3) lies on the line segment joining the points \[A\left( - \frac{2}{5}, 6 \right)\] and B (2, 8), find the value of m.
Write the perimeter of the triangle formed by the points O (0, 0), A (a, 0) and B (0, b).
Find the values of x for which the distance between the point P(2, −3), and Q (x, 5) is 10.
If P (x, 6) is the mid-point of the line segment joining A (6, 5) and B (4, y), find y.
The coordinates of a point on x-axis which lies on the perpendicular bisector of the line segment joining the points (7, 6) and (−3, 4) are
Point P(– 4, 2) lies on the line segment joining the points A(– 4, 6) and B(– 4, – 6).
Find the coordinates of the point whose ordinate is – 4 and which lies on y-axis.
Assertion (A): The ratio in which the line segment joining (2, -3) and (5, 6) internally divided by x-axis is 1:2.
Reason (R): as formula for the internal division is `((mx_2 + nx_1)/(m + n) , (my_2 + ny_1)/(m + n))`
