Advertisements
Advertisements
Question
The points A(–1, –2), B(4, 3), C(2, 5) and D(–3, 0) in that order form a rectangle.
Options
True
False
Advertisements
Solution
This statement is True.
Explanation:
Distance between A(–1, –2), B(4, 3),
AB = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2`
AB = `sqrt((4 + 1)^2 + (3 + 2)^2`
= `sqrt(5^2 + 5^2)`
= `sqrt(25 + 25)`
= `5sqrt(2)`
Distance between C(2, 5) and D(–3, 0),
CD = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2`
CD = `sqrt((-3 - 2)^2 + (0 - 5)^2`
= `sqrt((-5)^2 + (-5)^2)`
= `sqrt(25 + 25)`
= `5sqrt(2)`
Distance between A(–1, –2) and D(–3, 0),
AD = `sqrt((-3 + 1)^2 + (0 + 2)^2`
= `sqrt((-2)^2 + 2^2)`
= `sqrt(4 + 4)`
= `2sqrt(2)`
And distance between B(4, 3) and C(2, 5),
BC = `sqrt((4 - 2)^2 + (3 - 5)^2`
= `sqrt(2^2 + (-2)^2)`
= `sqrt(4 + 4)`
= `2sqrt(2)`
We know that, in a rectangle, opposite sides and equal diagonals are equal and bisect each other.
Since, AB = CD and AD = BC
Also, distance between A(–1, –2) and C(2, 5),
AC = `sqrt((2 + 1)^2 + (5 + 2)^2`
= `sqrt(3^2 + 7^2)`
= `sqrt(9 + 49)`
= `sqrt(58)`
And distance between D(–3, 0) and B(4, 3),
DB = `sqrt((4 + 3)^2 + (3 - 0)^2`
= `sqrt(7^2 + 3^2)`
= `sqrt(49 + 9)`
= `sqrt(58)`
Since, diagonals AC and BD are equal.
Hence, the points A(–1, – 2), B(4, 3), C(2, 5) and D(–3, 0) form a rectangle.
APPEARS IN
RELATED QUESTIONS
If Q (0, 1) is equidistant from P (5, − 3) and R (x, 6), find the values of x. Also find the distance QR and PR.
If the point P(x, y ) is equidistant from the points A(5, 1) and B (1, 5), prove that x = y.
Find the centre of the circle passing through (6, -6), (3, -7) and (3, 3)
Given a triangle ABC in which A = (4, −4), B = (0, 5) and C = (5, 10). A point P lies on BC such that BP : PC = 3 : 2. Find the length of line segment AP.
For what values of k are the points (8, 1), (3, –2k) and (k, –5) collinear ?
Find the distances between the following point.
R(–3a, a), S(a, –2a)
Find the distance of the following point from the origin :
(6 , 8)
Find the distance between the following point :
(sec θ , tan θ) and (- tan θ , sec θ)
Prove that the points (6 , -1) , (5 , 8) and (1 , 3) are the vertices of an isosceles triangle.
A(2, 5), B(-2, 4) and C(-2, 6) are the vertices of a triangle ABC. Prove that ABC is an isosceles triangle.
Find the distance between the following pair of points:
`(sqrt(3)+1,1)` and `(0, sqrt(3))`
Find a point on the y-axis which is equidistant from the points (5, 2) and (-4, 3).
A point P lies on the x-axis and another point Q lies on the y-axis.
Write the abscissa of point Q.
Point P (2, -7) is the centre of a circle with radius 13 unit, PT is perpendicular to chord AB and T = (-2, -4); calculate the length of AB.
Calculate the distance between the points P (2, 2) and Q (5, 4) correct to three significant figures.
Find the distance of the following points from origin.
(a+b, a-b)
KM is a straight line of 13 units If K has the coordinate (2, 5) and M has the coordinates (x, – 7) find the possible value of x.
By using the distance formula prove that each of the following sets of points are the vertices of a right angled triangle.
(i) (6, 2), (3, -1) and (- 2, 4)
(ii) (-2, 2), (8, -2) and (-4, -3).
The distance between the point P(1, 4) and Q(4, 0) is ______.
|
Case Study Trigonometry in the form of triangulation forms the basis of navigation, whether it is by land, sea or air. GPS a radio navigation system helps to locate our position on earth with the help of satellites. |
- Make a labelled figure on the basis of the given information and calculate the distance of the boat from the foot of the observation tower.
- After 10 minutes, the guard observed that the boat was approaching the tower and its distance from tower is reduced by 240(`sqrt(3)` - 1) m. He immediately raised the alarm. What was the new angle of depression of the boat from the top of the observation tower?
