Advertisements
Advertisements
प्रश्न
Show that the points A (1, −2), B (3, 6), C (5, 10) and D (3, 2) are the vertices of a parallelogram.
Advertisements
उत्तर
The distance d between two points `(x_1,y_1) and (x_2, y_2)` is given by the formula
`d = sqrt((x_1 - x_2)^2 +(y_1 - y_2)^2)`
In a parallelogram the opposite sides are equal in length.
Here the four points are A(1, −2), B(3, 6), C(5, 10) and D(3, 2).
Let us check the length of the opposite sides of the quadrilateral that is formed by these points.
`AB = sqrt((1 - 3)^2 + (2 - 6))`
`= sqrt((-2)^2 + (-8)^2)`
`= sqrt(4 + 64)`
`AB = sqrt(68)`
`CD = sqrt((5 - 3)^2 + (10 - 2)^2)`
`= sqrt((2)^2 + (8)^2)`
`= sqrt(4 + 64)`
`CD = sqrt(68)`
We have one pair of opposite sides equal.
Now, let us check the other pair of opposite sides.
`BC = sqrt((3 - 5)^2 + (6 - 10)^2)`
`= sqrt((-2)^2 + (-4)^2)`
`=sqrt(4 + 16)`
`AD = sqrt20`
The other pair of opposite sides is also equal. So, the quadrilateral formed by these four points is definitely a parallelogram.
Hence we have proved that the quadrilateral formed by the given four points is a parallelogram
APPEARS IN
संबंधित प्रश्न
Name the type of quadrilateral formed, if any, by the following point, and give reasons for your answer:
(−3, 5), (3, 1), (0, 3), (−1, −4)
Find the distance between the following pair of points:
(a+b, b+c) and (a-b, c-b)
If the point P(x, y ) is equidistant from the points A(5, 1) and B (1, 5), prove that x = y.
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.
Find the distance between the points
P(a sin ∝,a cos ∝ )and Q( acos ∝ ,- asin ∝)
Find the distance of the following points from the origin:
(iii) C (-4,-6)
If P (x , y ) is equidistant from the points A (7,1) and B (3,5) find the relation between x and y
Find the distance between the following pair of points.
L(5, –8), M(–7, –3)
Determine whether the points are collinear.
P(–2, 3), Q(1, 2), R(4, 1)
Find the distances between the following point.
R(–3a, a), S(a, –2a)
If the point P(2, 1) lies on the line segment joining points A(4, 2) and B(8, 4), then ______.
Find the distance of a point (7 , 5) from another point on the x - axis whose abscissa is -5.
A point P lies on the x-axis and another point Q lies on the y-axis.
If the abscissa of point P is -12 and the ordinate of point Q is -16; calculate the length of line segment PQ.
The centre of a circle is (2x - 1, 3x + 1). Find x if the circle passes through (-3, -1) and the length of its diameter is 20 unit.
Calculate the distance between A (5, -3) and B on the y-axis whose ordinate is 9.
Find distance of point A(6, 8) from origin
Find distance CD where C(– 3a, a), D(a, – 2a)
The point which lies on the perpendicular bisector of the line segment joining the points A(–2, –5) and B(2, 5) is ______.
∆ABC with vertices A(–2, 0), B(2, 0) and C(0, 2) is similar to ∆DEF with vertices D(–4, 0), E(4, 0) and F(0, 4).
|
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?
