Advertisements
Advertisements
प्रश्न
Prove that the points A(0, −1), B(−2, 3), C(6, 7) and D(8, 3) are the vertices of a rectangle ABCD?
उत्तर
Let the points (0, −1), (−2, 3), (6, 7), and (8, 3) be representing the vertices A, B, C, and D of a quadrilateral respectively.
Distance between the two points (x1, y1) and (x2, y2) is given by`sqrt((x_2-x_1)^2+(y_2-Y_1)^2)`
`thereforeAB=sqrt((0-(-2))^2+(-1-3)^2)``=sqrt((2)^2+(-4)^2)``=sqrt(4+16)=sqrt20=2sqrt5`
`BC=sqrt((-2-6)^2+(3-7)^2)=sqrt((-8)^2+(-4)^2)=sqrt64+16=sqrt80=4sqrt5`
`CD=sqrt((6-8)^2+(7-3)^2)=sqrt((-2)^2+(4)^2)=sqrt(4+16)=sqrt20=2sqrt5`
`AD=sqrt((8-0)^2+(3-(-1)^2))=sqrt((8)^2+(-8)^2)=sqrt64+16=sqrt80=4sqrt5`
Diagonal AC=`sqrt((0-6)^2+(-1-7)^2)=sqrt(6^2+(-8)^2)=sqrt100=10`
Diagonal AC=`sqrt((-2-8)^2+(3-3)^2)=sqrt((-10)^2+0^2)=sqrt100=10`
It can be observed that opposite sides of this quadrilateral are of the same length and also, the diagonals are of the same length.
Therefore, the given points are the vertices of a rectangle.
APPEARS IN
संबंधित प्रश्न
A ladder leaning against a wall makes an angle of 60° with the horizontal. If the foot of the ladder is 2.5 m away from the wall, find the length of the ladder
Nazima is fly fishing in a stream. The tip of her fishing rod is 1.8 m above the surface of the water and the fly at the end of the string rests on the water 3.6 m away and 2.4 m from a point directly under the tip of the rod. Assuming that her string (from the tip of her rod to the fly) is taut, ho much string does she have out (see Figure)? If she pulls in the string at the rate of 5 cm per second, what will be the horizontal distance of the fly from her after 12 seconds?
Identify, with reason, if the following is a Pythagorean triplet.
(4, 9, 12)
Identify, with reason, if the following is a Pythagorean triplet.
(24, 70, 74)
In the given figure, angle BAC = 90°, AC = 400 m, and AB = 300 m. Find the length of BC.
The top of a ladder of length 15 m reaches a window 9 m above the ground. What is the distance between the base of the wall and that of the ladder?
Each side of rhombus is 10cm. If one of its diagonals is 16cm, find the length of the other diagonals.
In a right-angled triangle ABC,ABC = 90°, AC = 10 cm, BC = 6 cm and BC produced to D such CD = 9 cm. Find the length of AD.
In a right-angled triangle PQR, right-angled at Q, S and T are points on PQ and QR respectively such as PT = SR = 13 cm, QT = 5 cm and PS = TR. Find the length of PQ and PS.
In the adjoining figure, a tangent is drawn to a circle of radius 4 cm and centre C, at the point S. Find the length of the tangent ST, if CT = 10 cm.
