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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)
उत्तर
Let the points (−3, 5), (3, 1), (0, 3), and (−1, −4) be representing the vertices A, B, C, and D of the given quadrilateral respectively.
AB = `sqrt((-3,-3)^2 + (5-1)^2)`
= `sqrt((-6)^2+(4)^2)`
= `sqrt(36+16)`
= `sqrt(52)`
= `2sqrt13`
BC = `sqrt((3-0)^2+(1-3)^2)`
= `sqrt((3)^2+(-2)^2)`
= `sqrt(9+4)`
= `sqrt13`
CD = `sqrt((0-(-1))^2+(3-(-4))^2)`
= `sqrt((1)^2+(7)^2)`
= `sqrt(1+49)`
= `sqrt50`
= `5sqrt2`
AD = `sqrt((-3-(-1))^2+(5-(-4))^2)`
= `sqrt((-2)^2+ (9)^2)`
= `sqrt(4+81)`
= `sqrt85`
AC = `sqrt ([0 - (-3)^2] + (3 - 5)^2)`
= `sqrt ((3)^2 + (-2)^2)`
= `sqrt (9 + 4)`
= `sqrt13`
BD = `sqrt ((-1 - 3)^2 + (-4 - 1)^1)`
= `sqrt ((-4)^2 + (5)^2)`
= `sqrt (16 + 25)`
= `sqrt41`
It can be observed that all sides of this quadrilateral are of different lengths. Therefore, it can be said that it is only a general quadrilateral, and not specific such as square, rectangle, etc.
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संबंधित प्रश्न
Determine if the points (1, 5), (2, 3) and (−2, −11) are collinear.
If A (-1, 3), B (1, -1) and C (5, 1) are the vertices of a triangle ABC, find the length of the median through A.
Find the distance between the points
A(-6,-4) and B(9,-12)
Using the distance formula, show that the given points are collinear:
(6, 9), (0, 1) and (-6, -7)
Show that the ▢PQRS formed by P(2, 1), Q(–1, 3), R(–5, –3) and S(–2, –5) is a rectangle.
Distance of point (-3, 4) from the origin is .....
(A) 7 (B) 1 (C) 5 (D) 4
Find the distance of the following point from the origin :
(13 , 0)
P and Q are two points lying on the x - axis and the y-axis respectively . Find the coordinates of P and Q if the difference between the abscissa of P and the ordinates of Q is 1 and PQ is 5 units.
Prove that the points (7 , 10) , (-2 , 5) and (3 , -4) are vertices of an isosceles right angled triangle.
Prove that the points (a, b), (a + 3, b + 4), (a − 1, b + 7) and (a − 4, b + 3) are the vertices of a parallelogram.
Find the distance between the following pairs of point:
`(sqrt(3)+1,1)` and `(0, sqrt(3))`
Find the co-ordinates of points on the x-axis which are at a distance of 17 units from the point (11, -8).
Give the relation that must exist between x and y so that (x, y) is equidistant from (6, -1) and (2, 3).
Find distance between point A(– 3, 4) and origin O
The distance of the point (α, β) from the origin is ______.
Case Study -2
A hockey field is the playing surface for the game of hockey. Historically, the game was played on natural turf (grass) but nowadays it is predominantly played on an artificial turf.
It is rectangular in shape - 100 yards by 60 yards. Goals consist of two upright posts placed equidistant from the centre of the backline, joined at the top by a horizontal crossbar. The inner edges of the posts must be 3.66 metres (4 yards) apart, and the lower edge of the crossbar must be 2.14 metres (7 feet) above the ground.
Each team plays with 11 players on the field during the game including the goalie. Positions you might play include -
- Forward: As shown by players A, B, C and D.
- Midfielders: As shown by players E, F and G.
- Fullbacks: As shown by players H, I and J.
- Goalie: As shown by player K.
Using the picture of a hockey field below, answer the questions that follow:
What are the coordinates of the position of a player Q such that his distance from K is twice his distance from E and K, Q and E are collinear?
Name the type of triangle formed by the points A(–5, 6), B(–4, –2) and C(7, 5).
Read the following passage:
|
Alia and Shagun are friends living on the same street in Patel Nagar. Shagun's house is at the intersection of one street with another street on which there is a library. They both study in the same school and that is not far from Shagun's house. Suppose the school is situated at the point O, i.e., the origin, Alia's house is at A. Shagun's house is at B and library is at C. |
Based on the above information, answer the following questions.
- How far is Alia's house from Shagun's house?
- How far is the library from Shagun's house?
- Show that for Shagun, school is farther compared to Alia's house and library.
OR
Show that Alia’s house, shagun’s house and library for an isosceles right triangle.
Read the following passage:
|
Use of mobile screen for long hours makes your eye sight weak and give you headaches. Children who are addicted to play "PUBG" can get easily stressed out. To raise social awareness about ill effects of playing PUBG, a school decided to start 'BAN PUBG' campaign, in which students are asked to prepare campaign board in the shape of a rectangle: One such campaign board made by class X student of the school is shown in the figure.
|
Based on the above information, answer the following questions:
- Find the coordinates of the point of intersection of diagonals AC and BD.
- Find the length of the diagonal AC.
-
- Find the area of the campaign Board ABCD.
OR - Find the ratio of the length of side AB to the length of the diagonal AC.
- Find the area of the campaign Board ABCD.
