Advertisements
Advertisements
Question
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
Solution
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
RELATED QUESTIONS
The diagonal of a quadrilateral shaped field is 24 m and the perpendiculars dropped on it from the remaining opposite vertices are 8 m and 13 m. Find the area of the field.
Find the area of a quadrilateral one of whose diagonals is 30 cm long and the perpendiculars from the other two vertices are 19 cm and 11 cm respectively.
A rectangular plot of land measures 45 m x 30 m. A boundary wall of height 2.4 m is built all around the plot at a distance of 1 m from the plot. Find the area of the inner surface of the boundary wall.
The diagonals of a quadrilateral are 16 cm and 13 cm. If they intersect each other at right angles; find the area of the quadrilateral.
The area of a rectangular is 640 m2. Taking its length as x cm; find in terms of x, the width of the rectangle. If the perimeter of the rectangle is 104 m; find its dimensions.
Calculate the area of quadrilateral ABCD, in which ∠ABD = 90°, triangle BCD is an equilateral triangle of side 24 cm and AD = 26 cm.
Calculate the area of quadrilateral ABCD in which AB = 32 cm, AD = 24 cm ∠A = 90° and BC = CD = 52 cm.
The perimeter of a rectangular field is `3/5`km. If the length of the field is twice its width; find the area of the rectangle in sq. meters.
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.
Two adjacent sides of a parallelogram are 24 cm and 18 cm. If the distance between the longer sides is 12 cm; find the distance between the shorter sides.
