Advertisements
Advertisements
प्रश्न
PQRS is a rectangle formed by joining the points P(– 1, – 1), Q(– 1, 4), R(5, 4) and S(5, – 1). A, B, C and D are the mid-points of PQ, QR, RS and SP respectively. Is the quadrilateral ABCD a square, a rectangle or a rhombus? Justify your answer.
Advertisements
उत्तर
Mid point of a line = `((x_1 + x_2)/2, (y_1 + y_2)/2)`
Mid point of PQ (A) = `((-1 - 1)/2, (-1 + 4)/2)`
= `((-2)/2, 3/2)`
= `(-1, 3/2)`
Mid point of QR (B) = `((-1 + 5)/2, (4 + 4)/2) = (4/2, 8/2)` = (2, 4)
Mid point of RS (C) =`((5 + 5)/2, (4 - 1)/2) = (10/2, 3/2) = (5, 3/2)`
Mid point of PS (D) = `((5 - 1)/2, (-1 - 1)/2) = (4/2, (-2)/2)` = (2, − 1)
Distance = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2`
AB = `sqrt((2 + 1)^2 + (4 - 3/2)^2`
= `sqrt(3^2 + (5/2)^2`
= `sqrt(9 + 25/4)`
= `sqrt((36 + 25)/4`
= `sqrt(61/4)`
BC = `sqrt((5 - 2)^2 + (3/2 - 4)^2`
= `sqrt(9 + 25/4)`
= `sqrt((36 + 25)/4`
= `sqrt(61/4)`
CD = `sqrt((5 - 2)^2 + (3/2 + 1)^2`
= `sqrt(3^2 + (5/2)^2`
= `sqrt(9 + 25/4)`
= `sqrt((36 + 25)/4)`
= `sqrt(61/4)`
AD = `sqrt((2 + 1)^2 + (-1 - 3/2)^2`
= `sqrt(3^2 + (- 5/2)^2`
= `sqrt(9 + 25/4)`
= `sqrt(61/4)`
AB = BC = CD = AD = `sqrt(61/4)`
Since all the four sides are equal,
∴ ABCD is a rhombus.
APPEARS IN
संबंधित प्रश्न
The diagonal of a rectangular plot is 34 m and its perimeter is 92 m. Find its area.
The perimeter of a rectangular board is 70 cm. Taking its length as x cm, find its width in terms of x.
If the area of the rectangular board is 300 cm2; find its dimensions.
Trapezium given below; find its area.
How many tiles, each of area 400 cm2, will be needed to pave a footpath which is 2 m wide and surrounds a grass plot 25 m long and 13 m wide?
The cost of enclosing a rectangular garden with a fence all around, at the rate of 75 paise per metre, is Rs. 300. If the length of the garden is 120 metres, find the area of the field in square metres.
The length of a rectangular verandah is 3 m more than its breadth. The numerical value of its area is equal to the numerical value of its perimeter.
(i) Taking x as the breadth of the verandah, write an equation in x that represents the above statement
(ii) Solve the equation obtained in (i) above and hence find the dimensions of the verandah.
The area of a rhombus is 216 sq. cm. If it's one diagonal is 24 cm; find:
(i) Length of its other diagonal,
(ii) Length of its side,
(iii) The perimeter of the rhombus.
A quadrilateral field of unequal has a longer diagonal with 140m. The perpendiculars from opposite vertives upon this diagonal are 20m and 14m. Find the area of the field.
Find the area of quadrilateral PQRS.
Vertices of given triangles are taken in order and their areas are provided aside. Find the value of ‘p’.
| Vertices | Area (sq.units) |
| (p, p), (5, 6), (5, –2) | 32 |
